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# Calculus with Applications

## Educators

JN
ka
+ 4 more educators

### Problem 1

Convert the following degree measures to radians. Leave answers as multiples of $\pi$.
$$65^{\circ}$$

Sherrie F.

### Problem 2

Convert the following degree measures to radians. Leave answers as multiples of $\pi .$
$$90^{\circ}$$

Sid W.
University of Louisville

### Problem 3

Convert the following degree measures to radians. Leave answers as multiples of $\pi$.
$$95^{\circ}$$

Sherrie F.

### Problem 4

Convert the following degree measures to radians. Leave answers as multiples of $\pi .$
$$135^{\circ}$$

Sid W.
University of Louisville

### Problem 5

Convert the following degree measures to radians. Leave answers as multiples of $\pi .$
$$270^{\circ}$$

Sherrie F.

### Problem 6

Convert the following degree measures to radians. Leave answers as multiples of $\pi .$
$$320^{\circ}$$

Sid W.
University of Louisville

### Problem 7

Convert the following degree measures to radians. Leave answers as multiples of $\pi .$
$$495^{\circ}$$

JN
Joel N.

### Problem 8

Convert the following degree measures to radians. Leave answers as multiples of $\pi .$
$$510^{\circ}$$

Sid W.
University of Louisville

### Problem 9

Convert the following radian measures to degrees.
$$\frac{5 \pi}{4}$$

Check back soon!

### Problem 10

Convert the following radian measures to degrees.
$$\frac{7 \pi}{6}$$

Sid W.
University of Louisville

### Problem 11

Convert the following radian measures to degrees.
$$-\frac{13 \pi}{6}$$

Nathan C.

### Problem 12

Convert the following radian measures to degrees.
$$-\frac{9 \pi}{4}$$

Sid W.
University of Louisville

### Problem 13

Convert the following radian measures to degrees.
$$\frac{8 \pi}{5}$$

Sherrie F.

### Problem 14

Convert the following radian measures to degrees.
$$\frac{9 \pi}{5}$$

Sid W.
University of Louisville

### Problem 15

Convert the following radian measures to degrees.
$$\frac{7 \pi}{12}$$

ka
Kanika A.

### Problem 16

Convert the following radian measures to degrees.
$$5 \pi$$

Sid W.
University of Louisville

### Problem 17

Find the values of the six trigonometric functions for the angles in standard position having the points in Exercises $17-20$ on their terminal sides.
$$(-3,4)$$

Sherrie F.

### Problem 18

Find the values of the six trigonometric functions for the angles in standard position having the points in Exercises $17-20$ on their terminal sides.
$$(-12,-5)$$

Sid W.
University of Louisville

### Problem 19

Find the values of the six trigonometric functions for the angles in standard position having the points in Exercises $17-20$ on their terminal sides.
$$(7,-24)$$

Sherrie F.

### Problem 20

Find the values of the six trigonometric functions for the angles in standard position having the points in Exercises $17-20$ on their terminal sides.
$$(20,15)$$

Sid W.
University of Louisville

### Problem 21

In quadrant $I, x, y,$ and $r$ are all positive, so that all six trigonometric functions have positive values. In quadrant II, $x$ is negative and $y$ is positive $(r$ is always positive). Thus, in quadrant II, sine is positive, cosine is negative, and so on. For Exercises $21-24,$ complete the following table of values for the signs of the trigonometric functions.

Sherrie F.

### Problem 22

In quadrant $I, x, y,$ and $r$ are all positive, so that all six trigonometric functions have positive values. In quadrant II, $x$ is negative and $y$ is positive $(r$ is always positive). Thus, in quadrant II, sine is positive, cosine is negative, and so on. For Exercises $21-24,$ complete the following table of values for the signs of the trigonometric functions.

Sid W.
University of Louisville

### Problem 23

In quadrant $I, x, y,$ and $r$ are all positive, so that all six trigonometric functions have positive values. In quadrant II, $x$ is negative and $y$ is positive $(r$ is always positive). Thus, in quadrant II, sine is positive, cosine is negative, and so on. For Exercises $21-24,$ complete the following table of values for the signs of the trigonometric functions.

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### Problem 24

In quadrant $I, x, y,$ and $r$ are all positive, so that all six trigonometric functions have positive values. In quadrant II, $x$ is negative and $y$ is positive $(r$ is always positive). Thus, in quadrant II, sine is positive, cosine is negative, and so on. For Exercises $21-24,$ complete the following table of values for the signs of the trigonometric functions.

Sid W.
University of Louisville

### Problem 25

For Exercises $25-32,$ complete the following table. Use the $30^{\circ}-60^{\circ}-90^{\circ}$ and $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Do not use a calculator.
$30^{\circ} \quad 1 / 2 \quad \sqrt{3} / 2 \quad \sqrt{3} / 2$

Sherrie F.

### Problem 26

For Exercises $25-32$ , complete the following table. Use the $30^{\circ}-60^{\circ}-90^{\circ}$ and $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Do not use a calculator.
$45^{\circ}$

Sid W.
University of Louisville

### Problem 27

For Exercises $25-32$ , complete the following table. Use the $30^{\circ}-60^{\circ}-90^{\circ}$ and $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Do not use a calculator.
$60^{\circ}$

Sherrie F.

### Problem 28

For Exercises $25-32$ , complete the following table. Use the $30^{\circ}-60^{\circ}-90^{\circ}$ and $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Do not use a calculator.
$120^{\circ}$

Sid W.
University of Louisville

### Problem 29

For Exercises $25-32$ , complete the following table. Use the $30^{\circ}-60^{\circ}-90^{\circ}$ and $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Do not use a calculator.
$135^{\circ}$

Sherrie F.

### Problem 30

For Exercises $25-32$ , complete the following table. Use the $30^{\circ}-60^{\circ}-90^{\circ}$ and $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Do not use a calculator.
$150^{\circ}$

Sid W.
University of Louisville

### Problem 31

For Exercises $25-32$ , complete the following table. Use the $30^{\circ}-60^{\circ}-90^{\circ}$ and $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Do not use a calculator.
$210^{\circ}$

Sherrie F.

### Problem 32

For Exercises $25-32$ , complete the following table. Use the $30^{\circ}-60^{\circ}-90^{\circ}$ and $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Do not use a calculator.
$240^{\circ}$

Sid W.
University of Louisville

### Problem 33

Find the following function values without using a calculator.
$$\sin \frac{\pi}{3}$$

Sherrie F.

### Problem 34

Find the following function values without using a calculator.
$$\cos \frac{\pi}{6}$$

Sid W.
University of Louisville

### Problem 35

Find the following function values without using a calculator.
$$\tan \frac{\pi}{4}$$

Sherrie F.

### Problem 36

Find the following function values without using a calculator.
$$\cot \frac{\pi}{3}$$

Sid W.
University of Louisville

### Problem 37

Find the following function values without using a calculator.
$$\csc \frac{\pi}{6}$$

Jonathan P.

### Problem 38

Find the following function values without using a calculator.
$$\sin \frac{3 \pi}{2}$$

Sid W.
University of Louisville

### Problem 39

Find the following function values without using a calculator.
$$\cos 3 \pi$$

Sherrie F.

### Problem 40

Find the following function values without using a calculator.
$$\sec \pi$$

Sid W.
University of Louisville

### Problem 41

Find the following function values without using a calculator.
$$\sin \frac{7 \pi}{4}$$

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### Problem 42

Find the following function values without using a calculator.
$$\tan \frac{5 \pi}{2}$$

Sid W.
University of Louisville

### Problem 43

Find the following function values without using a calculator.
$$\sec \frac{5 \pi}{4}$$

Sherrie F.

### Problem 44

Find the following function values without using a calculator.
$$\cos 5 \pi$$

Sid W.
University of Louisville

### Problem 45

Find the following function values without using a calculator.
$$\cot -\frac{3 \pi}{4}$$

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### Problem 46

Find the following function values without using a calculator.
$$\tan -\frac{5 \pi}{6}$$

Sid W.
University of Louisville

### Problem 47

Find the following function values without using a calculator.
$$\sin -\frac{7 \pi}{6}$$

Sherrie F.

### Problem 48

Find the following function values without using a calculator.
$$\cos -\frac{\pi}{6}$$

Sid W.
University of Louisville

### Problem 49

Find all values of $\theta$ between 0 and 2$\pi$ that satisfy each of the following equations.
$$\cos \theta=1 / 2$$

Sherrie F.

### Problem 50

Find all values of $\theta$ between 0 and 2$\pi$ that satisfy each of the following equations.
$$\sin \theta=-1 / 2$$

Sid W.
University of Louisville

### Problem 51

Find all values of $\theta$ between 0 and 2$\pi$ that satisfy each of the following equations.
$$\tan \theta=-1$$

Sherrie F.

### Problem 52

Find all values of $\theta$ between 0 and 2$\pi$ that satisfy each of the following equations.
$$\tan \theta=\sqrt{3}$$

Sid W.
University of Louisville

### Problem 53

Find all values of $\theta$ between 0 and 2$\pi$ that satisfy each of the following equations.
$$\sec \theta=-2 / \sqrt{3}$$

Sherrie F.

### Problem 54

Find all values of $\theta$ between 0 and 2$\pi$ that satisfy each of the following equations.
$$\sec \theta=\sqrt{2}$$

Sid W.
University of Louisville

### Problem 55

Use a calculator to find the following function values.
$$\sin 39^{\circ}$$

Christine A.

### Problem 56

Use a calculator to find the following function values.
$$\cos 67^{\circ}$$

Sid W.
University of Louisville

### Problem 57

Use a calculator to find the following function values.
$$\tan 123^{\circ}$$

Sherrie F.

### Problem 58

Use a calculator to find the following function values.
$$\tan 54^{\circ}$$

Sid W.
University of Louisville

### Problem 59

Use a calculator to find the following function values.
$$\sin 0.3638$$

Sherrie F.

### Problem 60

Use a calculator to find the following function values.
$$\tan 1.0123$$

Sid W.
University of Louisville

### Problem 61

Use a calculator to find the following function values.
$$\cos 1.2353$$

Sherrie F.

### Problem 62

Use a calculator to find the following function values.
$$\sin 1.5359$$

Sid W.
University of Louisville

### Problem 63

Find the amplitude $(a)$ and period $(T)$ of each function.
$$f(x)=\cos (3 x)$$

Sherrie F.

### Problem 64

Find the amplitude $(a)$ and period $(T)$ of each function.
$$h(x)=-\frac{1}{2} \sin (4 \pi x)$$

Sid W.
University of Louisville

### Problem 65

Find the amplitude $(a)$ and period $(T)$ of each function.
$$g(t)=-2 \sin \left(\frac{\pi}{4} t+2\right)$$

Sherrie F.

### Problem 66

Find the amplitude $(a)$ and period $(T)$ of each function.
$$s(t)=3 \sin (880 \pi t-7)$$

Sid W.
University of Louisville

### Problem 67

Graph each function defined as follows over a two-period interval.
$$y=2 \cos x$$

Sherrie F.

### Problem 68

Graph each function defined as follows over a two-period interval.
$$y=2 \sin x$$

Sid W.
University of Louisville

### Problem 69

Graph each function defined as follows over a two-period interval.
$$y=-\frac{1}{2} \cos x$$

Sherrie F.

### Problem 70

Graph each function defined as follows over a two-period interval.
$$y=-\sin x$$

Sid W.
University of Louisville

### Problem 71

Graph each function defined as follows over a two-period interval.
$$y=4 \sin \left(\frac{1}{2} x+\pi\right)+2$$

Sherrie F.

### Problem 72

Graph each function defined as follows over a two-period interval.
$$y=2 \cos \left(3 x-\frac{\pi}{4}\right)+1$$

Sid W.
University of Louisville

### Problem 73

Graph each function defined as follows over a two-period interval.
$$y=-3 \tan x$$

KZ
Kiska Z.

### Problem 74

Graph each function defined as follows over a two-period interval.
$$y=\frac{1}{2} \tan x$$

Sid W.
University of Louisville

### Problem 75

Consider the triangle shown on the next page, in which the three angles $\theta$ are equal and all sides have length $2 .$
(a) Using the fact that the sum of the angles in a triangle is $180^{\circ}$ , what are the measures of the three equal angles $\theta$ ?
(b) Suppose the triangle is cut in half as shown by a vertical line. What are the measures of the angles in the blue tri- angle on the left?
(c) What are the measures of the sides of the blue triangle on the left? (Hint: Once you've found the length of

Sherrie F.

### Problem 76

Consider the right triangle shown, in which the two sides have length $1 .$
(a) Using the Pythagorean Theorem, what is the length of the hypotenuse?
(b) Using the fact that the sum of the angles in a triangle is $180^{\circ}$ , what are the measures of the three angles?

Sid W.
University of Louisville

### Problem 77

Sales Sales of snowblowers are seasonal. Suppose the sales of snowblowers in one region of the country are approximated by
$$S(t)=500+500 \cos \left(\frac{\pi}{6} t\right)$$
where $t$ is time in months, with $t=0$ corresponding to November. Find the sales for $(a)-(e)$
(a) November $\quad$ (b) January $\quad$ (c) February (d) May (e) August $\quad$ (f) Graph $y=S(t)$

Sherrie F.

### Problem 78

APPLY IT Electricity Consumption The amount of electricity (in trillion BTUs) consumed by U.S. residential customers in 2013 is given in the following table. Source: Energy Information Administration.
(a) Plot the data, letting $t=1$ correspond to January, $t=2$ to February, and so on. Is it reasonable to assume that electri- cal consumption is periodic?
(b) Use a calculator with trigonometric regression to find a trigonometric function of the form
$$C(t)=a \sin (b t+c)+d$$
that models these data when $t$ is the month and $C(t)$ is the amount of electricity consumed (in trillion BTUs). Graph the function on the same calculator window as the data.
(c) Determine the period, $T,$ of the function found in part (b). Discuss the reasonableness of this period.
(d) Use the function from part (b) to estimate the consumption for the month of October, and compare it to the actual value.

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### Problem 79

Transylvania Hypothesis The "Transylvania hypothesis" claims that the full moon has an effect on health-related behavior. A study investigating this effect found a significant relationship between the phase of the moon and the number of general practice consultations nationwide, given by
$$y=100+1.8 \cos \left[\frac{(t-6) \pi}{14.77}\right],$$
where $y$ is the number of consultations as a percentage of the daily mean and $t$ is the days since the last full moon. Source: Family Practice.
(a) What is the period of this function? What is the significance of this period?
(b) There was a full moon on October $8,2014 .$ On what day in October 2014 does this formula predict the maximum number of consultations? What percent increase would be predicted for that day?
(c) What does the formula predict for October $25,2014 ?$

Sherrie F.

### Problem 80

Hyperkalemia Diagnosis A person with Hyperkalemia is monitored on an ECG (electrocardiogram). The graph shows that he has an inter-beat interval (conventionally named RR interval) of 32 seconds, an amplitude of 1.23 volts, a vertical shift of 2.4 volts, and a horizontal shift of 5 seconds.
(a) Find an equation giving the voltage of electricity as a function of time in seconds.
(b) After how many seconds does the electricity reaches its maximum amplitude?
(c) What is the value of voltage after 2 minutes?

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### Problem 81

Alzheimer's Disease A study on the circadian rhythms of patients with Alzheimer's disease found the body temperature of patients could be described by a function of the form
$$T=T_{0}+a \cos \left(\frac{2 \pi(t-k)}{24}\right),$$
where $t$ is the time in hours since midnight. For the patients without Alzheimer's, the average values of $T_{0}$ (the MESOR), $a$ (the amplitude), and $k$ (the acrophase) were $36.91^{\circ} \mathrm{C}, 0.32^{\circ} \mathrm{C},$ and 14.92 hours, while for the patients with the disease, the values were $37.29^{\circ} \mathrm{C}$ $0.46^{\circ} \mathrm{C},$ and 16.37 hours. Source: American Journal of Psychiatry.
(a) Graph the functions giving the temperature for each of the two groups using a graphing calculator. Do these two functions ever cross?
(b) At what time is the temperature highest for the patients without Alzheimer's?
(c) At what time is the temperature highest for the patients with Alzheimer's?

Sherrie F.

### Problem 82

Air Pollution The amount of pollution in the air fluctuates with the seasons. It is lower after heavy spring rains and higher after periods of little rain. In addition to this seasonal fluctuation, the long-term trend in many areas is upward. An idealized graph of this situation is shown in the figure below. Trigonometric functions can be used to describe the fluctuating part of the pollution levels. Powers of the number e can be used to show the long-term growth. In fact, the pollution level in a
certain area might be given by
$$P(t)=7(1-\cos 2 \pi t)(t+10)+100 e^{0.2 t}$$
where $t$ is time in years, with $t=0$ representing January 1 of the base year. Thus, July 1 of the same year would be represented by $t=0.5,$ while October 1 of the following year would be represented by $t=1.75 .$ Find the pollution levels on the following dates.
$\begin{array}{ll}{\text { (a) January } 1, \text { base year }} & {\text { (b) July } 1, \text { base year }} \\ {\text { (c) January } 1, \text { following year }} & {\text { (d) July } 1, \text { following year }}\end{array}$

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### Problem 83

Air Pollution Using a computer or a graphing calculator, sketch the function for air pollution given in Exercise 82 over the interval $[0,6] .$

Sherrie F.

### Problem 84

In Exercises 84 and $85,$ assume that $c_{1}=3 \times 10^{8} m$ per second, and find the speed of light in the second medium.
$$\theta_{1}=39^{\circ}, \theta_{2}=28^{\circ}$$

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### Problem 85

In Exercises 84 and $85,$ assume that $c_{1}=3 \times 10^{8} m$ per second, and find the speed of light in the second medium.
$$\theta_{1}=46^{\circ}, \theta_{2}=31^{\circ}$$

Sherrie F.

### Problem 86

Sound Pure sounds produce single sine waves on an oscilloscope. Find the period of each sine wave in the photographs in Exercises 86 and $87 .$ On the vertical scale each square represents $0.5,$ and on the horizontal scale each square represents $30^{\circ} .$

Sid W.
University of Louisville

### Problem 87

Sound Pure sounds produce single sine waves on an oscilloscope. Find the period of each sine wave in the photographs in Exercises 86 and $87 .$ On the vertical scale each square represents $0.5,$ and on the horizontal scale each square represents $30^{\circ} .$

Sherrie F.

### Problem 88

Sound Suppose the A key above Middle $\mathrm{C}$ is played as a pure tone. For this tone,
$$P(t)=0.002 \sin (880 \pi t)$$
where $P(t)$ is the change of pressure (in pounds per square foot) on a person's eardrum at time $t$ (in seconds). Source: The Physics and Psychophysics of Music: An Introduction.
(a) Graph this function on $[0,0.003]$ .
(b) Determine analytically the values of $t$ for which $P=0$ on $[0,0.003]$ and check graphically.
(c) Determine the period $T$ of $P(t)$ and the frequency of the A note.

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### Problem 89

Temperature The maximum afternoon temperature degrees Fahrenheit) in a given city is approximated by
$$T(t)=60-30 \cos (t / 2)$$
where $t$ represents the month, with $t=0$ representing January, $t=1$ representing February, and so on. Use a calculator to find the maximum afternoon temperature for the following months.
(a) Februry $\quad$ (b) April $\quad$ (c) September (d) July (e) December

Sherrie F.

### Problem 90

Temperature $\quad$ A mathematical model for the temperature in Fairbanks is
$$T(t)=37 \sin \left[\frac{2 \pi}{365}(t-101)\right]+25$$
where $T(t)$ is the temperature (in degrees Fahrenheit) on day $t$ with $t=0$ corresponding to January 1 and $t=364$ corresponding to December $31 .$ Use a calculator to estimate the tem- perature for $(a)-(d) .$ Source: The Mathematics Teacher.
(a) March 16$($ Day 74$)$ (b) May 2 (Day 121$)$ (c) Day 250 (d) Day 325 (e) Find maximum and minimum values of $T .$ (f) Find the period, T.

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### Problem 91

Sunset The number of minutes after noon, Eastern Standard Time, that the sun sets in Boston for specific days of the year is approximated in the following table. Source: The Old Farmer's Almanac.
(a) Plot the data. Is it reasonable to assume that the times of sunset are periodic?
(b) Use a calculator with trigonometric regression to find a trigonometric function of the form $s(t)=a \sin (b t+c)+d$ that models these data when $t$ is the day of the year and $s(t)$ is the number of minutes past noon, Eastern Standard Time that the sun sets
(c) Estimate the time of sunset for days $60,120,240 .$ Round answers to the nearest minute. (Hint: Don't forget about daylight savings time.)
(d) Use part (b) to estimate the days of the year that the sun sets at $6 : 00$ p.M. In reality, the days are close to 82 and $290 .$

Sherrie F.

### Problem 92

Cameras In the Kodak Customer Service Pamphlet $A A-26$ , Optical Formulas and Their Applications, the near and far limits of the depth of field (how close or how far away an object can be placed and still be in focus) are given by
$$w_{1}=\frac{u^{2}(\tan \theta)}{L+u(\tan \theta)} \quad and \quad \mathrm{w}_{2}=\frac{u^{2}(\tan \theta)}{L-u(\tan \theta)}$$
In these equations, $\theta$ represents the angle between the lens and the "circle of confusion," which is the circular image on the film of a point that is not exactly in focus. (The pamphlet suggests letting $\theta=1 / 30^{\circ} .$ ) $L$ is the diameter of the lens opening, which is found by dividing the focal length by the f-stop. (This is camera jargon you need not worry about here.) For this problem, let the focal length be $50 \mathrm{mm},$ or $0.05 \mathrm{m} ;$ if the lens is set at $\mathrm{f} / 8,$ then $L=0.05 / 8=0.00625 \mathrm{m} .$ Finally, $u$ is the distance to the object being photographed. Find the near and far limits of the depth of field when the object being photographed is 6 $\mathrm{m}$ from the camera.

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### Problem 93

Measurement A surveyor standing 65 $\mathrm{m}$ from the base of a building measures the angle to the top of the building and finds it to be $42.8^{\circ} .$ (See the figure.) Use trigonometry to find the height of the building.

Sherrie F.

### Problem 94

Measurement Jenny Crum stands on a cliff at the edge of a canyon. On the opposite side of the canyon is another cliff equal in height to the one she is on. (See the figure.) By dropping a rock and timing its fall, she determines that it is 105 ft to the bottom of the canyon. She also determines that the angle to the base of the opposite cliff is $27^{\circ} .$ How far is it to the opposite side of the canyon?

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### Problem 95

Whitewater Rafting A mathematics textbook author rafting down the Colorado River was told by a guide that the river dropped an average of 26 ft per mile as it ran through Cataract Canyon. Find the average angle of the river with the horizontal in degrees. Hint: Find the tangent of the angle, and then use a calculator to find the angle where the tangent has that value. There are 5280 ft in a mile. Be sure your calculator is set on degrees.)

Sherrie F.

### Problem 96

Computer Drawing A mathematics professor wanted to use a computer drawing program to draw a picture of a regular pentagon (a five-sided figure with sides of equal length and with equal angles). He first made a 1 -in. base by drawing a line from $(0,0)$ to $(1,0) .$ (See the figure.) He then needed to find the coordinates of the other three vertex points. Use trigonometry to find them. (Hint: The sum of the exterior angles of any polygon is $360^{\circ} . )$

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### Problem 97

Amusement Rides A proud father is attempting to take a picture of his daughters while they are riding on a merry-go-round. Horses on this particular ride move up and down as the ride progresses according to the function
$$h(t)=\sin \left(\frac{t}{\pi}-2\right)+4$$
where $h(t)$ represents the height (in feet) of the horse's nose at time $t,$ relative to the merry-go-round platform. However, because of safety fencing surrounding the ride, it is only possible to get a good picture when the height of the horse's nose is between 3.5 and 4 ft off the merry-go-round platform. Find the first time interval that the father has to take the picture.

Sherrie F.