## Chapter Review Exercises

### Solve Quadratic Equations Using the Square Root Property

Exercise (PageIndex{1}) Solve Quadratic Equations of the Form (ax^{2}=k) Using the Square Root Property

In the following exercises, solve using the Square Root Property.

- (y^{2}=144)
- (n^{2}-80=0)
- (4 a^{2}=100)
- (2 b^{2}=72)
- (r^{2}+32=0)
- (t^{2}+18=0)
- (frac{2}{3} w^{2}-20=30)
- (5 c^{2}+3=19)

**Answer**1. (y=pm 12)

3. (a=pm 5)

5. (r=pm 4 sqrt{2} i)

7. (w=pm 5 sqrt{3})

Exercise (PageIndex{2}) Solve Quadratic Equations of the Form (a(x-h)^{2}=k) Using the Square Root Property

In the following exercises, solve using the Square Root Property.

- ((p-5)^{2}+3=19)
- ((u+1)^{2}=45)
- (left(x-frac{1}{4} ight)^{2}=frac{3}{16})
- (left(y-frac{2}{3} ight)^{2}=frac{2}{9})
- ((n-4)^{2}-50=150)
- ((4 c-1)^{2}=-18)
- (n^{2}+10 n+25=12)
- (64 a^{2}+48 a+9=81)

**Answer**1. (p=-1,9)

3. (x=frac{1}{4} pm frac{sqrt{3}}{4})

5. (n=4 pm 10 sqrt{2})

7. (n=-5 pm 2 sqrt{3})

### Solve Quadratic Equations by Completing the Square

Exercise (PageIndex{3}) Solve Quadratic Equations Using Completing the Square

In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

- (x^{2}+22 x)
- (m^{2}-8 m)
- (a^{2}-3 a)
- (b^{2}+13 b)

**Answer**1. ((x+11)^{2})

3. (left(a-frac{3}{2} ight)^{2})

Exercise (PageIndex{4}) Solve Quadratic Equations Using Completing the Square

In the following exercises, solve by completing the square.

- (d^{2}+14 d=-13)
- (y^{2}-6 y=36)
- (m^{2}+6 m=-109)
- (t^{2}-12 t=-40)
- (v^{2}-14 v=-31)
- (w^{2}-20 w=100)
- (m^{2}+10 m-4=-13)
- (n^{2}-6 n+11=34)
- (a^{2}=3 a+8)
- (b^{2}=11 b-5)
- ((u+8)(u+4)=14)
- ((z-10)(z+2)=28)

**Answer**1. (d=-13,-1)

3. (m=-3 pm 10 i)

5. (v=7 pm 3 sqrt{2})

7. (m=-9,-1)

9. (a=frac{3}{2} pm frac{sqrt{41}}{2})

11. (u=-6 pm 2 sqrt{2})

### Solve Quadratic Equations of the Form (ax^{2}+bx+c=0) by Completing the Square

Exercise (PageIndex{5}) Solve Quadratic Equations of the Form (ax^{2}+bx+c=0) by Completing the Square

In the following exercises, solve by completing the square.

- (3 p^{2}-18 p+15=15)
- (5 q^{2}+70 q+20=0)
- (4 y^{2}-6 y=4)
- (2 x^{2}+2 x=4)
- (3 c^{2}+2 c=9)
- (4 d^{2}-2 d=8)
- (2 x^{2}+6 x=-5)
- (2 x^{2}+4 x=-5)

**Answer**1. (p=0,6)

3. (y=-frac{1}{2}, 2)

5. (c=-frac{1}{3} pm frac{2 sqrt{7}}{3})

7. (x=frac{3}{2} pm frac{1}{2} i)

Exercise (PageIndex{6}) Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

- (4 x^{2}-5 x+1=0)
- (7 y^{2}+4 y-3=0)
- (r^{2}-r-42=0)
- (t^{2}+13 t+22=0)
- (4 v^{2}+v-5=0)
- (2 w^{2}+9 w+2=0)
- (3 m^{2}+8 m+2=0)
- (5 n^{2}+2 n-1=0)
- (6 a^{2}-5 a+2=0)
- (4 b^{2}-b+8=0)
- (u(u-10)+3=0)
- (5 z(z-2)=3)
- (frac{1}{8} p^{2}-frac{1}{5} p=-frac{1}{20})
- (frac{2}{5} q^{2}+frac{3}{10} q=frac{1}{10})
- (4 c^{2}+4 c+1=0)
- (9 d^{2}-12 d=-4)

**Answer**1. (x=frac{1}{4}, 1)

3. (r=-6,7)

5. (v=frac{-1 pm sqrt{21}}{8})

7. (m=frac{-4 pm sqrt{10}}{3})

9. (a=frac{5}{12} pm frac{sqrt{23}}{12} i)

11. (u=5 pm sqrt{21})

13. (p=frac{4 pm sqrt{5}}{5})

15. (c=-frac{1}{2})

Exercise (PageIndex{7}) Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation

In the following exercises, determine the number of solutions for each quadratic equation.

- (9 x^{2}-6 x+1=0)
- (3 y^{2}-8 y+1=0)
- (7 m^{2}+12 m+4=0)
- (5 n^{2}-n+1=0)

- (5 x^{2}-7 x-8=0)
- (7 x^{2}-10 x+5=0)
- (25 x^{2}-90 x+81=0)
- (15 x^{2}-8 x+4=0)

**Answer**1.

- (1)
- (2)
- (2)
- (2)

Exercise (PageIndex{8}) Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

- (16 r^{2}-8 r+1=0)
- (5 t^{2}-8 t+3=9)
- (3(c+2)^{2}=15)

- (4 d^{2}+10 d-5=21)
- (25 x^{2}-60 x+36=0)
- (6(5 v-7)^{2}=150)

**Answer**1.

- Factor
- Quadratic Formula
- Square Root

### Solve Equations in Quadratic Form

Exercise (PageIndex{9}) Solve Equations in Quadratic Form

In the following exercises, solve.

- (x^{4}-14 x^{2}+24=0)
- (x^{4}+4 x^{2}-32=0)
- (4 x^{4}-5 x^{2}+1=0)
- ((2 y+3)^{2}+3(2 y+3)-28=0)
- (x+3 sqrt{x}-28=0)
- (6 x+5 sqrt{x}-6=0)
- (x^{frac{2}{3}}-10 x^{frac{1}{3}}+24=0)
- (x+7 x^{frac{1}{2}}+6=0)
- (8 x^{-2}-2 x^{-1}-3=0)

**Answer**1. (x=pm sqrt{2}, x=pm 2 sqrt{3})

3. (x=pm 1, x=pm frac{1}{2})

5. (x=16)

7. (x=64, x=216)

9. (x=-2, x=frac{4}{3})

### Solve Applications of Quadratic Equations

Exercise (PageIndex{10}) Solve Applications Modeled by Quadratic Equations

In the following exercises, solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

- Find two consecutive odd numbers whose product is (323).
- Find two consecutive even numbers whose product is (624).
- A triangular banner has an area of (351) square centimeters. The length of the base is two centimeters longer than four times the height. Find the height and length of the base.
- Julius built a triangular display case for his coin collection. The height of the display case is six inches less than twice the width of the base. The area of the of the back of the case is (70) square inches. Find the height and width of the case.
- A tile mosaic in the shape of a right triangle is used as the corner of a rectangular pathway. The hypotenuse of the mosaic is (5) feet. One side of the mosaic is twice as long as the other side. What are the lengths of the sides? Round to the nearest tenth.

*Figure 9.E.1*

6. A rectangular piece of plywood has a diagonal which measures two feet more than the width. The length of the plywood is twice the width. What is the length of the plywood’s diagonal? Round to the nearest tenth.

7. The front walk from the street to Pam’s house has an area of (250) square feet. Its length is two less than four times its width. Find the length and width of the sidewalk. Round to the nearest tenth.

8. For Sophia’s graduation party, several tables of the same width will be arranged end to end to give serving table with a total area of (75) square feet. The total length of the tables will be two more than three times the width. Find the length and width of the serving table so Sophia can purchase the correct size tablecloth . Round answer to the nearest tenth.

9. A ball is thrown vertically in the air with a velocity of (160) ft/sec. Use the formula (h=-16 t^{2}+v_{0} t) to determine when the ball will be (384) feet from the ground. Round to the nearest tenth.

10. The couple took a small airplane for a quick flight up to the wine country for a romantic dinner and then returned home. The plane flew a total of (5) hours and each way the trip was (360) miles. If the plane was flying at (150) mph, what was the speed of the wind that affected the plane?

11. Ezra kayaked up the river and then back in a total time of (6) hours. The trip was (4) miles each way and the current was difficult. If Roy kayaked at a speed of (5) mph, what was the speed of the current?

12. Two handymen can do a home repair in (2) hours if they work together. One of the men takes (3) hours more than the other man to finish the job by himself. How long does it take for each handyman to do the home repair individually?

**Answer**2. Two consecutive even numbers whose product is (624) are (24) and (26), and (−24) and (−26).

4. The height is (14) inches and the width is (10) inches.

6. The length of the diagonal is (3.6) feet.

8. The width of the serving table is (4.7) feet and the length is (16.1) feet.

10. The speed of the wind was (30) mph.

12. One man takes (3) hours and the other man (6) hours to finish the repair alone.

### Graph Quadratic Functions Using Properties

Exercise (PageIndex{11}) Recognize the Graph of a Quadratic Function

In the following exercises, graph by plotting point.

- Graph (y=x^{2}-2)
- Graph (y=-x^{2}+3)

**Answer**2.

Exercise (PageIndex{12}) Recognize the Graph of a Quadratic Function

In the following exercises, determine if the following parabolas open up or down.

- (y=-3 x^{2}+3 x-1)
- (y=5 x^{2}+6 x+3)

- (y=x^{2}+8 x-1)
- (y=-4 x^{2}-7 x+1)

**Answer**2.

- Up
- Down

Exercise (PageIndex{13}) Find the Axis of Symmetry and Vertex of a Parabola

In the following exercises, find

- The equation of the axis of symmetry
- The vertex
- (y=-x^{2}+6 x+8)
- (y=2 x^{2}-8 x+1)

**Answer**2. (x=2) ; ((2,-7))

Exercise (PageIndex{14}) Find the Intercepts of a Parabola

In the following exercises, find the (x)- and (y)-intercepts.

- (y=x^{2}-4x+5)
- (y=x^{2}-8x+15)
- (y=x^{2}-4x+10)
- (y=-5x^{2}-30x-46)
- (y=16x^{2}-8x+1)
- (y=x^{2}+16x+64)

**Answer**2. (egin{array}{l}{y :(0,15)} {x :(3,0),(5,0)}end{array})

4. (egin{array}{l}{y :(0,-46)} {x : ext { none }}end{array})

6. (egin{array}{l}{y :(0,-64)} {x :(-8,0)}end{array})

#### Graph Quadratic Functions Using Properties

Exercise (PageIndex{15}) Graph Quadratic Functions Using Properties

In the following exercises, graph by using its properties.

- (y=x^{2}+8 x+15)
- (y=x^{2}-2 x-3)
- (y=-x^{2}+8 x-16)
- (y=4 x^{2}-4 x+1)
- (y=x^{2}+6 x+13)
- (y=-2 x^{2}-8 x-12)

**Answer**2.

4.

6.

Exercise (PageIndex{16}) Solve Maximum and Minimum Applications

In the following exercises, find the minimum or maximum value.

- (y=7 x^{2}+14 x+6)
- (y=-3 x^{2}+12 x-10)

**Answer**2. The maximum value is (2) when (x=2).

Exercise (PageIndex{17}) Solve Maximum and Minimum Applications

In the following exercises, solve. Rounding answers to the nearest tenth.

- A ball is thrown upward from the ground with an initial velocity of (112) ft/sec. Use the quadratic equation (h=-16 t^{2}+112 t) to find how long it will take the ball to reach maximum height, and then find the maximum height.
- A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using (180) feet of fencing on three sides of the yard. The quadratic equation (A=-2 x^{2}+180 x) gives the area, (A), of the yard for the length, (x), of the building that will border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximum area.

**Answer**2. The length adjacent to the building is (90) feet giving a maximum area of (4,050) square feet.

### Graph Quadratic Functions Using Transformations

Exercise (PageIndex{18}) Graph Quadratic Functions of the Form (f(x)=x^{2}+k)

In the following exercises, graph each function using a vertical shift.

- (g(x)=x^{2}+4)
- (h(x)=x^{2}-3)

**Answer**2.

Exercise (PageIndex{19}) Graph Quadratic Functions of the Form (f(x)=x^{2}+k)

In the following exercises, graph each function using a horizontal shift.

- (f(x)=(x+1)^{2})
- (g(x)=(x-3)^{2})

**Answer**2.

Exercise (PageIndex{20}) Graph Quadratic Functions of the Form (f(x)=x^{2}+k)

In the following exercises, graph each function using transformations.

- (f(x)=(x+2)^{2}+3)
- (f(x)=(x+3)^{2}-2)
- (f(x)=(x-1)^{2}+4)
- (f(x)=(x-4)^{2}-3)

**Answer**2.

4.

Exercise (PageIndex{21}) Graph Quadratic Functions of the Form (f(x)=ax^{2})

In the following exercises, graph each function.

- (f(x)=2x^{2})
- (f(x)=-x^{2})
- (f(x)=frac{1}{2} x^{2})

**Answer**2.

Exercise (PageIndex{22}) Graph Quadratic Functions Using Transformations

In the following exercises, rewrite each function in the (f(x)=a(x-h)^{2}+k) form by completing the square.

- (f(x)=2 x^{2}-4 x-4)
- (f(x)=3 x^{2}+12 x+8)

**Answer**1. (f(x)=2(x-1)^{2}-6)

Exercise (PageIndex{23}) Graph Quadratic Functions Using Transformations

In the following exercises,

- Rewrite each function in (f(x)=a(x−h)^{2}+k) form
- Graph it by using transformations
- (f(x)=3 x^{2}-6 x-1)
- (f(x)=-2 x^{2}-12 x-5)
- (f(x)=2 x^{2}+4 x+6)
- (f(x)=3 x^{2}-12 x+7)

**Answer**1.

- (f(x)=3(x-1)^{2}-4)
*Figure 9.E.13*

3.

- (f(x)=2(x+1)^{2}+4)
*Figure 9.E.14*

Exercise (PageIndex{24}) Graph Quadratic Functions Using Transformations

In the following exercises,

- Rewrite each function in (f(x)=a(x−h)^{2}+k) form
- Graph it using properties
- (f(x)=-3 x^{2}-12 x-5)
- (f(x)=2 x^{2}-12 x+7)

**Answer**1.

- (f(x)=-3(x+2)^{2}+7)
*Figure 9.E.15*

Exercise (PageIndex{25}) Find a Quadratic Function From its Graph

In the following exercises, write the quadratic function in (f(x)=a(x−h)^{2}+k) form.

*Figure 9.E.16**Figure 9.E.17*

**Answer**1. (f(x)=(x+1)^{2}-5)

### Solve Quadratic Inequalities

Exercise (PageIndex{26}) Solve Quadratic Inequalities Graphically

In the following exercises, solve graphically and write the solution in interval notation.

- (x^{2}-x-6>0)
- (x^{2}+4 x+3 leq 0)
- (-x^{2}-x+2 geq 0)
- (-x^{2}+2 x+3<0)

**Answer**1.

*Figure 9.E.18*- ((-infty,-2) cup(3, infty))

3.

*Figure 9.E.19*- ([-2,1])

Exercise (PageIndex{27}) Solve Quadratic Inequalities Graphically

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

- (x^{2}-6 x+8<0)
- (x^{2}+x>12)
- (x^{2}-6 x+4 leq 0)
- (2 x^{2}+7 x-4>0)
- (-x^{2}+x-6>0)
- (x^{2}-2 x+4 geq 0)

**Answer**1. ((2,4))

3. ([3-sqrt{5}, 3+sqrt{5}])

5. no solution

## Practice Test

Exercise (PageIndex{28})

- Use the Square Root Property to solve the quadratic equation (3(w+5)^{2}=27).
- Use Completing the Square to solve the quadratic equation (a^{2}-8 a+7=23).
- Use the Quadratic Formula to solve the quadratic equation (2 m^{2}-5 m+3=0).

**Answer**1. (w=-2, w=-8)

3. (m=1, m=frac{3}{2})

Exercise (PageIndex{29})

Solve the following quadratic equations. Use any method.

- (2 x(3 x-2)-1=0)
- (frac{9}{4} y^{2}-3 y+1=0)

**Answer**2. (y=frac{2}{3})

Exercise (PageIndex{30})

Use the discriminant to determine the number and type of solutions of each quadratic equation.

- (6 p^{2}-13 p+7=0)
- (3 q^{2}-10 q+12=0)

**Answer**2. (2) complex

Exercise (PageIndex{31})

Solve each equation.

- (4 x^{4}-17 x^{2}+4=0)
- (y^{frac{2}{3}}+2 y^{frac{1}{3}}-3=0)

**Answer**2. (y=1, y=-27)

Exercise (PageIndex{32})

For each parabola, find

- Which direction it opens
- The equation of the axis of symmetry
- The vertex
- The (x)
*-*and (y)-intercepts - The maximum or minimum value
- (y=3 x^{2}+6 x+8)
- (y=-x^{2}-8 x+16)

**Answer**2.

- down
- (x=-4)
- ((-4,0))
- (y: (0,16); x: (-4,0))
- minimum value of (-4) when (x=0)

Exercise (PageIndex{33})

Graph each quadratic function using intercepts, the vertex, and the equation of the axis of symmetry.

- (f(x)=x^{2}+6 x+9)
- (f(x)=-2 x^{2}+8 x+4)

**Answer**2.

Exercise (PageIndex{34})

In the following exercises, graph each function using transformations.

- (f(x)=(x+3)^{2}+2)
- (f(x)=x^{2}-4 x-1)

**Answer**2.

*Figure 9.E.21*

Exercise (PageIndex{35})

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

- (x^{2}-6 x-8 leq 0)
- (2 x^{2}+x-10>0)

**Answer**2. (left(-infty,-frac{5}{2} ight) cup(2, infty))

Exercise (PageIndex{36})

Model the situation with a quadratic equation and solve by any method.

- Find two consecutive even numbers whose product is (360).
- The length of a diagonal of a rectangle is three more than the width. The length of the rectangle is three times the width. Find the length of the diagonal. (Round to the nearest tenth.)

**Answer**2. A water balloon is launched upward at the rate of (86) ft/sec. Using the formula (h=-16 t^{2}+86 t) find how long it will take the balloon to reach the maximum height, and then find the maximum height. Round to the nearest tenth.

## 19.E: Review Exercises 2

**Answer Keys to H. Hansen and G. Quinn, Greek: An Intensive Course. 2nd revised ed. Fordham UP, 1992.**

I created these keys while teaching ancient Greek for the first time at Duke University during the 2003&ndash2004 academic year. Since then I have used them as student aids in Summer Greek courses at the University of Arizona. I am posting them here in the hopes that they will prove useful for students encountering Hansen and Quinn for the first time, whether in a classroom or on their own. They are intended to provide guidance to those with questions about the material, but not to help cheaters. **If you are a student submitting information you find here as your own work, you are plagiarizing. Plagiarizing is bad! Your instructor will likely not approve.**

As even the casual user will discover, t he keys have their limitations. For instance, my translations into Greek and English generally represent only one way of interpreting originals, though sometimes I have included alternate versions placed in (parentheses) or __underlined__. In addition, in most cases more literal English translations have been preferred to those which sound better. As a result, many of the translations of the Readings are awkward (though usually accurate). Finally, it should be noted that though the keys have been corrected a number of times (and more often than not by Petra Axolotl, to whom I owe a great debt of gratitude), errors doubtless remain. So use with caution! And if you find a mistake, please contact me.

## Aquatic exercise for adults with type 2 diabetes: a meta-analysis

**Aims:** The purpose of this systematic review and meta-analysis was to examine the effects of aquatic exercise (AquaEx) on indicators of glycemic control (i.e., glycated hemoglobin [A1c] and fasting plasma glucose) in adults with type 2 diabetes mellitus (T2DM). It was hypothesized that AquaEx would improve glycemic control to a similar extent as land-based exercise (LandEx), but to a greater extent than non-exercise control (Ctrl).

**Methods:** A literature search was completed in February 2017 for studies examining AquaEx training in adults with T2DM. Assessment of glycemic control was necessary for inclusion, while secondary outcomes such as quality of life and cardiometabolic risk factors (i.e., blood pressure, triglycerides and total cholesterol) were considered, but not required for inclusion. Outcomes were measured before and after at least 8 weeks of AquaEx, and data were analyzed using weighted mean differences (WMDs) and fixed effect models, when appropriate.

**Results:** Nine trials including 222 participants were identified. Three trials compared AquaEx to LandEx, two compared AquaEx to Crtl, and four had a pre-/post-design without a comparison group. Results indicate no difference in A1c between LandEx and AquaEx (WMD = -0.02%, 95% confidence interval = [-0.71, 0.66]). Post-intervention A1c was lower in AquaEx when compared to Crtl (WMD = -0.96%, [-1.87, -0.05]). Post-AquaEx A1c was lower compared to baseline (WMD = -0.48%, [-0.66, -0.30]).

**Conclusions:** A1c can be reduced after eight-twelve weeks of AquaEx. However, at this time few studies have examined whether changes in A1c are different from LandEx or Crtl.

**Keywords:** Aquatic exercise Glycated hemoglobin Swimming Type 2 diabetes mellitus.

## Exercises for mechanical neck disorders: A Cochrane review update

**Background:** Neck pain (NP) is disabling and costly.

**Objectives:** To assess the effectiveness of exercise on pain, disability, function, patient satisfaction, quality of life (QoL) and global perceived effect (GPE) in adults with NP.

**Methods:** We searched computerised databases up to May 2014 for randomized controlled trials (RCTs) comparing exercise to a control in adults with NP with/without cervicogenic headache (CGH) or radiculopathy. Two reviewers independently conducted selection, data abstraction and assessed risk of bias. Meta-analyses were performed to establish pooled standardised mean differences (SMDp). The Grade of Recommendation, Assessment, Development and Evaluation (GRADE) was used to summarise the body of evidence.

**Main results:** The following exercises (27 trials) were supported by 'Moderate GRADE' evidence: For chronic NP, 1) cervico-scapulothoracic and upper extremity (UE) strengthening for moderate to large pain reduction immediately post treatment (IP) and at short-term (ST) follow-up 2) scapulothoracic and UE endurance training for a small pain reduction (IP/ST) 3) cervical, shoulder and scapulothoracic strengthening and stretching exercise for a small to large pain reduction in the long-term (LT) (SMDp -0.45 [95%CI: -0.72 to -0.18]) and function improvement 4) cervico-scapulothoracic strengthening/stabilisation exercises for pain and function at intermediate-term (IT) (SMDp -14.90 [95%CI: -22.40 to -7.39]). 5) mindfulness exercises (Qigong) for minor improved function but not GPE (ST). For chronic CGH, cervico-scapulothoracic strengthening and endurance exercises including pressure biofeedback for small/moderate improvement of pain, function and GPE (IP/LT).

**Authors' conclusions:** Specific strengthening exercises of the neck, scapulothoracic and shoulder for chronic NP and chronic CGH are beneficial. Future research should explore optimal dosage.

**Keywords:** Cochrane review Exercise Meta-analysis Neck pain.

## Numerical Problems

- Liquid nitrogen, which has a boiling point of &minus195.79°C, is used as a coolant and as a preservative for biological tissues. Is the entropy of nitrogen higher or lower at &minus200°C than at &minus190°C? Explain your answer. Liquid nitrogen freezes to a white solid at &minus210.00°C, with an enthalpy of fusion of 0.71 kJ/mol. What is its entropy of fusion? Is freezing biological tissue in liquid nitrogen an example of a reversible process or an irreversible process?
- Using the second law of thermodynamics, explain why heat flows from a hot body to a cold body but not from a cold body to a hot body.
- One test of the spontaneity of a reaction is whether the entropy of the universe increases: &DeltaS
_{univ}> 0. Using an entropic argument, show that the following reaction is spontaneous at 25°C:

Why does the entropy of the universe increase in this reaction even though gaseous molecules, which have a high entropy, are consumed?

Based on this table, can you conclude that entropy is related to the nature of functional groups? Explain your reasoning.

The text states that the magnitude of &DeltaS_{vap} tends to be similar for a wide variety of compounds. Based on the values in the table, do you agree?

## Sigma 19mm f/2.8 EX DN

Sigma announced that it would be providing lenses for mirrorless cameras, and the 19mm ƒ/2.8 DN ("Digital Neo") is the company's second lens to do so. The lens is offered in the micro four-thirds mount offered by Olympus and Panasonic, and the Sony E-mount for the NEX series of camera.

On a micro four-thirds camera body, the lens will offer an equivalent field of view of 38mm on a Sony NEX camera body, the lens offers an equivalent field of view of just over 28mm (both of these figures are in the 35mm film equivalent).

The lens ships with a round lens hood, takes 46mm filters, and is available now for around $200.

**Sharpness**

The lens provides sharp images, more so on the micro four-thirds camera than on the Sony NEX, owing to the fact that the micro four-thirds sensor does not "see" the corners of the lens.

Mounted on the Panasonic GX-1 m4/3 camera, the lens produces excellently sharp images even wide open at ƒ/2.8. There's just a hint of corner softness at ƒ/2.8, but even this is all but eliminated by just stopping down to ƒ/4. Stopping down to ƒ/5.6 technically provides more sharpness, but you'd only notice it looking very closely at a test chart. Diffraction limiting sets in at ƒ/8, but you won't notice any impact on sharpness until ƒ/16, where it is still very sharp across the frame. Fully stopped-down at ƒ/22, there is some impact on sharpness, but it's still very good.

Mounted on the Sony NEX-7, the lens follows the same pattern of sharpness - very good at ƒ/2.8, excellent at ƒ/4 to ƒ/8, and softer at ƒ/16 and ƒ/22. In this case however we note softer corners than seen on the GX-1, owing to the larger sensor being viewed by the lens.

**Chromatic Aberration**

Chromatic aberration is notable with the lens attached to both camera bodies: on both bodies, it's viewed in the corners, in areas of high contrast. In images shot with the Panasonic GX-1, it appears as magenta-green fringing, and in images shot with the Sony NEX-7, it appears as magenta fringing. It gets marginally better with either camera if you stop down, but only slightly.

**Shading (''Vignetting'')**

With the Sigma 19mm ƒ/2.8 mounted on the Panasonic GX-1, corner shading isn't really an issue. However when the lens is mounted on the Sony NEX-7, there is always at least a slight amount of corner shading: at ƒ/2.8, the corners are almost a half-stop darker than the center stopped down, the corners are 1/3 EV darker than the center.

**Distortion**

Results for distortion testing with the Sigma 19mm ƒ/2.8 DN are the same with both cameras: there is some notable barrel distortion in the corners of the images (around +0.5%).

**Autofocus Operation**

The Sigma 19mm ƒ/2.8 DN uses an in-lens motor for focus and is very fast to do so: focusing from infinity to close-focus took well less than one second. There is no extension of the lens during focus operations and attached filters will not rotate.

**Macro**

This is not a lens with any significant macro capability: it only offers 0.14x magnification, with a minimum close-focusing distance of just under 8 inches (20cm).

**Build Quality and Handling**

The Sigma 19mm ƒ/2.8 EX DN is a fairly pedestrian lens, small and light with a matte black finish. The lens is made of primarily plastic components, weighing only 140 grams (4.9oz), with plastic 46mm filter threads and a metal lens mount. The lens has only one control on it, the manual focus ring. There is no depth-of-field scale, distance scale or infrared index.

The focusing ring is about 3/4'' wide, with deep, easy-to-grip plastic ribs. There are no stops, hard or otherwise, at the close-focusing and infinity end of the lens.

The included lens hood is plastic, attaches via a bayonet mount, and can be reversed onto the lens for storage. The interior of the hood is ribbed, and when attached to the lens it adds about 3/4'' to the overall length.

**Alternatives**

$250

For the Sony users, the Sigma 19mm ƒ/2.8 offers a slightly longer alternative to the Sony 16mm ƒ/2.8. In this case however the Sigma also offers a sharper alternative, with the 19mm ƒ/2.8 offering significantly sharper results throughout the same apertures. Other test results are also better: CA is better handled, there is slightly less corner shading, and distortion is a bit less severe (naturally barrel-distorted rather than pincushion).

$400

At almost double the price of the Sigma 19mm, the Panasonic 20mm offers slightly sharper results with less CA. There is somewhat significant corner shading, but almost zero distortion.

$300

Olympus offers a wider alternative to the Sigma, which is about on the same level optically: however, we would give a nod to the Sigma for its better tolerance to chromatic aberration, and slightly sharper performance. In this case, the Olympus makes for a slightly smaller.

**Conclusion**

There are a surprising number of alternatives in this category - wide primes for mirrorless cameras - so Sigma's offering comes at an interesting time. It holds its own in our tests, providing a sharp image even wide open at ƒ/2.8, with only slightly notable chromatic aberration in the corners. It's not a case of Sigma finding a niche other manufacturers haven't exploited, however, Sigma does undercut all the current manufacturers with a lower price point. If you haven't got a wide angle prime for your micro four-thirds mirrorless camera, or your Sony NEX camera, then you may want to consider the Sigma 19mm ƒ/2.8 DN.

**Product Photos**

**Sample Photos**

The VFA target should give you a good idea of sharpness in the center and corners, as well as some idea of the extent of barrel or pincushion distortion and chromatic aberration, while the Still Life subject may help in judging contrast and color. We shoot both images using the default JPEG settings and manual white balance of our test bodies, so the images should be quite consistent from lens to lens.

As appropriate, we shoot these with both full-frame and sub-frame bodies, at a range of focal lengths, and at both maximum aperture and ƒ/8. For the ''VFA'' target (the viewfinder accuracy target from Imaging Resource), we also provide sample crops from the center and upper-left corner of each shot, so you can quickly get a sense of relative sharpness, without having to download and inspect the full-res images. To avoid space limitations with the layout of our review pages, indexes to the test shots launch in separate windows.

## Exercises to Help with Vertigo

Repetitive movements can help your brain and body overcome the confusing signals of vertigo. They can also help you manage the sudden onset of dizziness and motion sensations.

When you begin these exercises for vertigo, start slowly and understand that initial reactions may make you feel worse. Make sure that you complete these exercises individually, taking breaks between each one. Speak with your doctor before beginning any of these exercises, and let them know if your vertigo symptoms become more serious or if you experience any new symptoms.

**Brandt-Daroff Exercise**

This exercise helps to reduce the symptoms of vertigo, and it is most often used for BPPV and labyrinthitis.

Step 1: Sit at the end of your bed. When ready, turn your head 45 degrees to the right.

Step 2: Lie down on your left side and remain still for 30 seconds until dizziness fades.

Step 3: Sit up and wait 30 seconds.

Step 4: Turn your head 45 degrees to the left.

Step 5: Lie down on your right side, and hold the position for 30 seconds until dizziness fades.

Step 6: Sit up and wait 30 seconds.

Repeat this process five times, twice a day or as comfort allows.

**Marching in Place Exercise**

Marching in place can help you with balance while standing, and it acts as a stepping stone for more advanced movements.

Step 1: Stand near a wall or corner, or place a chair nearby. Place your arms by your side.

Step 2: Lift your right knee, followed by your left knee as you march. Try to raise your knees as high as comfort allows.

Step 3: March in place 20 times.

Repeat this exercise two times a day times, and try to extend each set to 30 marching steps.

**Turning in Place Exercise**

Turning in place is a more advanced exercise than marching in place. Make sure you have a chair or sturdy walker nearby in case you feel dizzy.

Step 1: Stand straight with your arms at your sides.

Step 2: Slowly turn left in a half-circle, equal to 180 degrees.

Step 3: Stop moving and stand motionless for 10 to 15 seconds.

Step 4: Slowly turn right in a half-circle. Stand still for 10 to 15 seconds.

Repeat this exercise five times. As you complete each round, favor moving in the direction that makes you feel dizzier.

**Epley Maneuver**

The Epley maneuver is one of two exercises, called canalith repositioning procedures, designed specifically for BPPV. Follow this exercise maneuver only if you are experiencing BPPV.

Step 1: Sit at the end of your bed and turn your head 45 degrees to the right.

Step 2: Maintain that position and lie back with head reclining and shoulders resting on a pillow. Hold for 30 seconds.

Step 3: Turn your head 90 degrees to the left and wait for 30 seconds.

Step 4: Turn your head and body 90 degrees to the left until you are face down on the bed. Hold for 30 seconds.

Step 5: Sit up on your left side.

These steps apply to the right ear. For left ear issues, reverse all directions:

Step 1: Sit at the end of your bed and turn your head 45 degrees to the left.

Step 2: Maintain that position and lie back with head reclining and shoulders resting on a pillow. Hold for 30 seconds.

Step 3: Turn your head 90 degrees to the right and wait for 30 seconds.

Step 4: Turn your head and body 90 degrees to the right until you are face down on the bed. Hold for 30 seconds.

Step 5: Sit up on your right side.

Repeat this exercise three times or as comfort allows.

**Semont Liberatory Maneuver**

The Semont Liberatory maneuver is the second exercise procedure for treating BPPV.

Step 1: Sit at the end of your bed and turn your head 45 degrees to the right.

Step 2: Lie down on your left side with your head tilted upright, and hold still for 60 seconds.

Step 3: In one motion, move from your left side to your right side. Make sure your face is facing the bed. Remain still for 60 seconds.

Step 4: Return to a sitting position and sit for 5 minutes.

These steps apply to the left ear. For right ear issues, reverse all directions:

Step 1: Sit at the end of a bed and turn your head 45 degrees to the left.

Step 2: Lie down on your right side with your head tilted upright, and hold still for 60 seconds.

Step 3: In one motion, move from your right side to your left side. Make sure your face is facing the bed. Remain still for 60 seconds.

Step 4: Return to a sitting position and sit for 5 minutes.

Repeat this exercise three times or as comfort allows.

## Pursed-Lips Breathing

Pursed-lips breathing is a breathing technique designed to make your breathing more effective by making the breaths slower and more intentional. After inhaling, you pucker your lips and exhale through them slowly and deliberately, often while counting.

Pursed-lips breathing has shown to be beneficial for people with anxiety that's associated with lung conditions, such as chronic obstructive pulmonary disease (COPD) and emphysema. It can be performed up to four to five times a day.

- Relax your neck and shoulders.
- Inhale slowly through the nostrils for 2 seconds (keep your mouth closed), a deep breath is unnecessary a normal breath will do just fine.
- Exhale through the mouth for 4 seconds (the extended time is the key). When exhaling, pucker your mouth as if giving a kiss.
- While breathing out, keep a slow and steady breath don’t breathe out hard.

## Sigma 19mm F2.8 DN A MFT mount review: Modest price, modest performance?

With an APS-C image circle, Sigma&rsquos DN series is specifically designed for mirrorless models from Sony and the smaller MFT format adopted by both Panasonic and Olympus. Originally launched as EX types with f2.8 apertures, the firm has refreshed the original two focal lengths (a wide and standard) designating them as their prestigious Art series and added a third, a 60mm short telephoto (reviewed previously).

As the wide-angle of the three, this 19mm A series lens is an upgrade of the earlier EX model of the same focal length and, with an MFT mount, is the equivalent to a 38mm f2.8. Although this model features a new cosmetic appearance with a thin metal skin and minimalist exterior, internally this lens features 8 elements (with no less than three aspheres) in 6 groups, like its predecessor.

Paradoxically, this 19mm model is larger than the 30mm model and measures 45.7mm in length (as opposed to 40.5mm of the 30mm focal length), focuses down to 20cm and weighs 160g. Although it lacks stabilization, the 19mm f2.8 DN A remains sensitively priced at $199.

On the Olympus OM-D E-M1 the Sigma 19mm f2.8 achieves a DxOMark score of 18 points. Although good it&rsquos not great putting it midway between the Lumix G 20mm f1.7 ASPH and Olympus M.Zuiko Digital 17mm f2.8 with DxOMark scores of 22 and 15 points, respectively. However, given those models&rsquo peak sharpness scores of 11P-Mpix and 6P-Mpix, the score of 7P-Mpix for the Sigma is on the low side. While center sharpness is best at full aperture it doesn&rsquot really improve when stopped down and has generally poor uniformity across the field. Not only that but lateral CA is a little higher than we would like with it being just visible in the periphery, sides and corners of the frame.

Compared with its predecessor the new model shows some marginal improvement in optical quality, particularly at full-aperture, however the gains are very minor. Transmission and control of chromatic aberration are also very slightly improved, and Sigma has also lowered the distortion, suggesting the firm has fine-tuned some of the optical components. However, in real world terms it would be difficult to tell.

While it&rsquos encouraging to see Sigma upgrading the quality of their lenses, much of that on this occasion is cosmetic. However, this model also shows some slight improvement in image quality, mostly at full aperture. Sharpness isn&rsquot everything though and at $199 it remains competitively priced when compared with rival offerings.

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## 20 Things From Amazon That'll Help You Exercise With Your Dog More

Exercise is equally important for both you and your dog. Yes, there’s always the standard daily walk, but there’s so much more out there if you’re looking to take your workouts to the next level.

We’ve found an at-home agility course, a doggie treadmill, a hands-free bicycle leash and more products that will help you and your dog reach your fitness goals (while still having fun). Check them out below.

*HuffPost may receive a share from purchases made via links on this page. Prices and availability subject to change.*

You can wear this around your waist. Let your pup set the pace for your walks, runs or hikes — or lead the way!

**Promising review:** "If you run, hike or bike with your dog regularly you NEED this leash in your life! It is the absolute best! My 3 y/o 80lb labradoodle is a very active boy and likes to go everywhere we go. It's hard to keep up with his exercise needs with my own two legs, **I wanted a leash that would allow me to ride my bike and this is great for that! Being hands-free is amazing.**" — Kate Mahoney

You'll have your dog running back and forth while giving your arm a killer workout. (Biceps? Triceps? All of the above.)

**Promising review:** "One of the best fetch toys out there. **Exercises your dog**, launches the ball far effortlessly, and can easily spot the orange ball. My 8-year-old boxer who is an obsessive-compulsive fetcher absolutely loves this." — Rintje

Use this to make your daily dog walks more exciting for you and your pup.

**Promising review:** "**These are wonderful for exercising your pup**. My dog (10 months) goes and gets it and begs me to play with him. It's fun and he gets lots of exercise running and jumping." — Juliet Hart

So your dog can join the fun of your neighborhood bike rides. It's suitable for all bike models and will keep your doggo safe with a stainless steel guide pole.

**Promising review:** "**This product is helping me get back in shape!** This bike leash has significantly reduced my dog's ability to knock me off my balance, which was a daily occurrence before. I highly recommend this leash." — DJ G.

You and your dog can start your own at-home FIFA tournament.

**Promising review:** "My dog loves to play ball especially with this because he can easily grab it and run. **He loves when you chase him around and try to take his ball.** Very well made." — Zoe

This will give your dog a full-body workout. You can get involved too — grab the other end for a game of tug of war!

**Promising review:** "My 8-month-old American bully loves this! Great installation instructions and extremely easy to set up! **He’ll be able to get a lot of exercise from this!**" — Lesa Danielson

Get you and your pups running in the yard, at the beach and everywhere in between.

**Promising review:** "We love this disc and a pack of five is perfect for endless throwing abilities! We have a high energy weim-lab mix who is obsessed with playing frisbee. **Good durability, wonderful visibility, and harm-free catching equals tons of fun for our dog and us!**" — Chelsie

Perfect for the tiny pups who want to join in on your adventures! Use this hiking, skiing, biking, walking or even dancing. The possibilities are endless!

**Promising review:** "Best decision I have ever made for my dogs, seriously life-changing! **Thanks to this pack, me and my dogs all get more exercise.** I tell people daily when I'm out walking the trails how much I love it!" — Jamie Thomsen

You'll be able to get your steps in early in the morning or at night when you have this.

**Promising review:** "I have two large dogs who need lots of exercise, and we often walk late at night or early morning. I bought one of these harnesses three years ago just to check it out. Loved it so much I immediately bought another. Whenever we walk in the dark, they are each wearing one. We've done this for three years constantly." — Watery M

These will level up your fetch game. These balls float and are light enough to throw long distances.

**Promising review:** "We have a lumbering Great Dane/Boxer and an energizer bunny Jack Russell/Border Collie. They both adore this ball! These are great for fetching, they float in water, they have a good bounce to them, and they hold up to a lot of intense chewing. **If you have a mouthy dog, a chewer, or a fetch fanatic, this is your toy.** Get several — you won't regret it!" — Amanda

You and your pup can become canoeing and kayaking voyagers with this.

**Promising review:** "I needed to get my dog in the local lake for some swimming exercise. This is easy to get on and off of him, adjust to size, keep him afloat, and priced within my budget. It gives you peace of mind when you are in or around the water with your best canine friend." — nanjking

Your pupper won't feel left out of your home workout sesh.

**Promising review:** "I love how compact the DogPacer is! With the side panels removed, you hardly notice it. It's relatively quiet and the track is plenty long enough for 50–65-lb dogs. Controls are simple and quick. Well worth the price!" — M.K.

With this, both you and your dog can get active outside. Build each other's agility by playing hide-and-seek, chase, fetch and more!

**Promising review:** "This is an awesome beginner agility set. You can set it up easily for the height you need and it is easy to put into the grass. I love that everything comes packed into two bags you can easily carry on your shoulder. They are light. Great set! Colorful and looks sturdy! **Plus, you get to be physically active and have fun with your dogs, while they get physical and mental exercise, which they need**!" — John L. Schieffer III

If you have a senior dog, you need this. The subtle wobble motion of these cushions helps to activate stabilizing muscles — great for rehabilitation or physical therapy!

**Promising review:** "The pet balance disc worked so well we ended up buying a second one! We really love it and so does our dog." — Jason M.

Your dog shouldn't be left out of the FitBit club. This collar tracks your dog's location and activity levels and also notifies you when they leave home and reach their goals.

**Promising review:** "Overall, l give this product the five stars. l was mostly interested in this as a tracker for a wayward pupper, but it does much more than this as it also tracks her fitness and general health. I would definitely purchase this item again." — Bruce E. Mitchell

This is lightweight, safe on your pets' teeth and perfect to take outside to get more active together.

**Promising review:** "Best toy ever! My dog LOVES this thing! It’s lightweight so he can toss it, run with it and it’s so durable we can even tug with it. Will be buying a few more just in case he loses it." — Amazon Customer

Protect your pup's feet on trickier climbing trails, walks and terrain of any kind. Plus, it's made of all-natural food-grade waxes and oils!

**Promising review:** "As the owner of a 60+ pound Catahoula who demands daily exercise the fact it's snowing or people who put rock salt down cannot deter me or him from our daily hour-long walks. So, the next best solution was that Musher's Wax. It's perfect. I cannot rate this product highly enough." — Shannon Lew

Fill this (with this weight even distributed on both sides), so your dog can feel involved with your backpacking, trekking or camping trip.

**Promising review:** "Great pack for my dog, I bought the large for my 90+ pound German shepherd. **Mainly bought it for exercise, add weight to the pockets on walks and during play time.** I like to take him on hikes with me, plenty of room for food, bags, water, treats etc. Overall great product, high quality material and stitching." — David Rubio

Give your dog the freedom to roam and wander without fear that they'll run away. You'll love using this on your walks to the dog park, camping or beach trips, hiking and even training!

**Promising review:** "My dog loves this leash! This leash will allow her to get way more exercise than our 30 minute walks do. We are also in the process of training her emergency recall and I’ll be using this leash for that as well. I haven’t had it long but I feel like it’s a good investment." —NavyChiefWife

With this, you and your pup can walk, run or bike longer without them getting dehydrated.

**Promising review:** "This water bottle has SAVED us on walks with our 4-month-old pup outside. She enjoys drinking out of this more than her own water bowl. SO convenient to carry around, leakproof and easy to clean! Highly recommend this product for all other pet humans." — Amanda Wang

## How to exercise safely with diabetes

Exercise is recommended for all people with diabetes, though some may have to take extra precautions. For example, people with type 1 diabetes should be particularly careful.

"For type 1 diabetics, exercise can lower blood sugar more dramatically," Hsu says. Dangerously low blood sugar, or hypoglycemia, can cause health complications including seizures and coma in severe cases.

People with type 1 diabetes should carefully plan their exercises around food intake and insulin dosage, according to the ADA. It's also important to measure your blood sugar levels before, during, and after exercise — or check your blood sugar with a continuous glucose monitor.

Overall, it's best to work with your doctor to develop a routine if you have type 1 diabetes.

If you have diabetes and are starting an exercise routine, you should take the following steps:

**Speak with your doctor.**Let them know if you've had any other health complications with diabetes, like eye problems, heart disease, or stroke.**Start slow.**Familiarize yourself with how exercise affects your blood sugar by measuring your blood sugar before and after exercise, and monitoring any major changes. Your blood sugar should stay within the healthy range that you and your doctor have established.**Monitor your feet for ulcers or sores**. Many diabetics have decreased sensation in their feet, Li says, so you might not notice pain from sores. Visual monitoring can help you spot them and prevent infection.