Question 1

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
-Suppose the tip of the minute hand of a clock is 3 inches from the center of the clock. Determine the distance traveled by the tip of the minute hand in $4.5$ hours. Give an exact answer.

Accepted Answer

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Question 2

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
-Find the radius (to the nearest hundredth of a millimeter) of a pulley if rotating the pulley $74.53 ^ { \circ }$ raises the pulley $16.0 \mathrm {~mm}$.

Accepted Answer

To find the radius of the pulley, we can use the relationship between the angle of rotation in radians, the radius of the pulley, and the arc length (which is the distance raised). The formula is $ \theta = \frac{s}{r} $, where $ \theta $ is the angle in radians, $ s $ is the arc length (distance raised), and $ r $ is the radius of the pulley.First, convert the angle from degrees to radians: $ 74.53^\circ = 74.53 \times \frac{\pi}{180} $ radians.Then, rearrange the formula to solve for $ r $: $ r = \frac{s}{\theta} $.Substitute $ s = 16.0 $ mm (the distance raised) and $ \theta = 74.53 \times \frac{\pi}{180} $ radians into the formula:$$ r = \frac{16.0}{74.53 \times \frac{\pi}{180}} $$$$ r \approx 12.30 $$ mmTherefore, the radius of the pulley is approximately 12.30 mm, making choice B correct.

Question 3

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
-The minute hand of a clock is 7 inches long. What distance does its tip move in 24 minutes? Give an exact answer.

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Question 4

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
-Suppose the tip of the minute hand of a clock is 9 inches from the center of the clock. Determine the distance traveled by the tip of the minute hand in 10 minutes. Give an exact answer.

Accepted Answer

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Question 5

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
-A bicycle with a 26-inch wheel (diameter) travels a distance of 300 feet. How many revolutions does the wheel make (to the nearest revolution)?

Accepted Answer

The distance traveled by the bicycle is 300 feet. To find the number of revolutions, we need to convert the wheel diameter from inches to feet, then calculate the circumference of the wheel, and finally divide the total distance by the wheel's circumference. The wheel's diameter is 26 inches, which is $26 \text{ inches} \times \frac{1 \text{ foot}}{12 \text{ inches}} = \frac{13}{6} \text{ feet}$. The circumference of the wheel is $\pi \times \text{diameter} = \pi \times \frac{13}{6} \text{ feet}$. The number of revolutions is the total distance divided by the circumference of the wheel, which is $300 \text{ feet} \div \left(\pi \times \frac{13}{6} \text{ feet}\right) \approx 44$ revolutions.

Question 6

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
-Suppose the tip of the minute hand of a clock is $2 \mathrm { in }$. from the center of the clock. Determine the distance traveled by the tip of the minute hand in $3 \frac { 1 } { 2 }$ hours. Give an exact answer.

Accepted Answer

The minute hand completes one full circle (circumference $4\pi$ in.) every hour. In $3 \frac{1}{2}$ hours, it travels $3.5 \times 4\pi = 14\pi$ inches.

Question 7

Find the area of a sector of a circle having radius r and central angle $$\theta$$ . If necessary, express the answer to the nearest
tenth.
-$$\mathrm { r } = 15.0 \mathrm {~m} , \theta = 20 ^ { \circ }$$

Accepted Answer

The area of a sector of a circle is given by the formula $\frac{1}{2} r^2 \theta$ where $r$ is the radius and $\theta$ is the central angle in radians. First, convert the angle from degrees to radians: $20^\circ = \frac{20\pi}{180} = \frac{\pi}{9}$ radians. Then, calculate the area: $\frac{1}{2} \times 15^2 \times \frac{\pi}{9} \approx 39.3 \mathrm{~m}^2$.

Question 8

Find the area of a sector of a circle having radius r and central angle $$\theta$$ . If necessary, express the answer to the nearest
tenth.
-$$\mathrm { r } = 7.0 \mathrm { mi } , \theta = 268 ^ { \circ }$$

Accepted Answer

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Question 9

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
-A pulley with a diameter of 28 inches is driven by a belt which is moving $1035 \mathrm { ft } / \mathrm { min }$. To the nearest unit, how many revolutions per minute are made by the pulley?

Accepted Answer

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Question 10

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
-Two wheels are rotating in such a way that the rotation of the smaller wheel causes the larger wheel to rotate. The radius of the smaller wheel is $6.6$ centimeters and the radius of the larger wheel is $15.2$ centimeters. Through how many degrees (to the nearest hundredth of a degree) will the larger wheel rotate if the smaller one rotates $38 ^ { \circ }$ ?

Accepted Answer

The rotation of the larger wheel is determined by the ratio of the radii of the wheels. The formula is: $(\text{degrees of smaller wheel}) \times \left(\frac{\text{radius of smaller wheel}}{\text{radius of larger wheel}}\right)$. Substituting the given values: $38^\circ \times \left(\frac{6.6}{15.2}\right) \approx 16.50^\circ$.

Question 11

Find the area of a sector of a circle having radius r and central angle $$\theta$$ . If necessary, express the answer to the nearest
tenth.
-$\mathrm { r } = 4.0 \mathrm { ft } , \theta = \frac { 2 \pi } { 3 }$ radians

Accepted Answer

The area of a sector is given by $\frac{1}{2} r^2 \theta$. Substituting $r = 4.0$ and $\theta = \frac{2\pi}{3}$, the area is $\frac{1}{2} \times 4^2 \times \frac{2\pi}{3} = \frac{1}{2} \times 16 \times \frac{2\pi}{3} = \frac{32\pi}{6} \approx 16.8 \mathrm{ft}^2$.

Question 12

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
-A wheel with a 32 -inch radius is marked at two points on the rim. The distance between the marks along the wheel is found to be 16 inches. What is the angle (to the nearest tenth of a degree) between the radii to the two marks?

Accepted Answer

The circumference of the earth is $2\pi r = 2\pi (6400) = 40212$ km. The distance between the marks on the wheel is 16 inches, or $\frac{16}{12} = \frac{4}{3}$ feet. The circumference of the wheel is $2\pi r = 2\pi (32/12) = \frac{32\pi}{6} = \frac{16\pi}{3}$ feet. The angle between the radii to the two marks is $\frac{4/3}{16\pi/3} = \frac{1}{4\pi} \approx 0.0796$ radians, or $0.0796 \cdot 180/\pi \approx 4.56^\circ$.

Question 13

Approximate the length using the formula for arc length. Round to the nearest meter.
-A television tower $420 \mathrm {~m}$ high subtends an angle of $4 ^ { \circ } 40 ^ { \prime }$. How far away is the tower?

Accepted Answer

To find the arc length, convert the angle from degrees and minutes to radians: $4^\circ 40' = 4 + \frac{40}{60} = 4.6667^\circ$. Convert to radians: $4.6667 \times \frac{\pi}{180} \approx 0.0814$ radians. Use the formula for arc length $s = r\theta$, where $r = 420 \mathrm{~m}$ and $\theta = 0.0814$. Thus, $s \approx 420 \times 0.0814 \approx 34.188$. The distance is the radius of the circle, so $r = \frac{420}{0.0814} \approx 5157 \mathrm{~m}$.

Question 14

Find the area of a sector of a circle having radius r and central angle $$\theta$$ . If necessary, express the answer to the nearest
tenth.
-$\mathrm { r } = 47.2 \mathrm {~cm} , \theta = \frac { \pi } { 11 }$ radians

Accepted Answer

The area of a sector is given by the formula $\frac{1}{2} r^2 \theta$. Substituting $r = 47.2$ cm and $\theta = \frac{\pi}{11}$, the area is $\frac{1}{2} \times (47.2)^2 \times \frac{\pi}{11} \approx 318.1 \mathrm{~cm}^2$.

Question 15

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
-A pulley rotates through $50 ^ { \circ }$ in one minute. How many rotations (to the nearest tenth of a rotation) does the pulley make in an hour?

Accepted Answer

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Question 16

Find the area of a sector of a circle having radius r and central angle $$\theta$$ . If necessary, express the answer to the nearest
tenth.
-$\frac { \pi } { 3 }$ Find the radius of a circle in which a central angle of radian determines a sector of area 57 square meters. Round to the nearest hundredth.

Accepted Answer

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Question 17

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
-A car wheel has a 14 -inch radius. Through what angle (to the nearest tenth of a degree) does the wheel turn when the car rolls forward $3 \mathrm { ft }$ ?

Accepted Answer

To find the angle, first convert the 3 feet distance to inches (36 inches). The circumference of the wheel is $2 \pi \times 14$. The angle in radians is $\frac{36}{2 \pi \times 14}$, which converts to degrees by multiplying by $\frac{180}{\pi}$, resulting in approximately $147.3^\circ$.

Question 18

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
-Suppose the tip of the minute hand of a clock is 6 inches from the center of the clock. Determine the distance traveled by the tip of the minute hand in 30 minutes. Give an exact answer.

Accepted Answer

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Question 19

Approximate the length using the formula for arc length. Round to the nearest meter.
-A tree $490 \mathrm {~m}$ away subtends an angle of $2 ^ { \circ }$. Find the height of the tree.

Accepted Answer

The arc length formula is $ s = r \theta $, where $ r $ is the radius and $ \theta $ is the angle in radians. Convert $2^\circ$ to radians: $ \theta = \frac{2\pi}{180} $. Then, $ s = 490 \times \frac{2\pi}{180} \approx 17 $ meters.

Question 20

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
-Electrical wire is being wound around a drum with radius of $1.10$ meters. How much line (to the nearest hundredth of a meter) would be wound around the drum if it is rotated through an angle of $270.1 ^ { \circ }$ ?

Accepted Answer

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Quiz 7: The Circular Functions and Their Graphs

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -Suppose the tip of the minute hand of a clock is 3 inches from the center of the clock. Determine the distance traveled by the tip of the minute hand in $4.5$ hours. Give an exact answer.

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -Find the radius (to the nearest hundredth of a millimeter) of a pulley if rotating the pulley $74.53 ^ { \circ }$ raises the pulley $16.0 \mathrm {~mm}$ .

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -The minute hand of a clock is 7 inches long. What distance does its tip move in 24 minutes? Give an exact answer.

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -Suppose the tip of the minute hand of a clock is 9 inches from the center of the clock. Determine the distance traveled by the tip of the minute hand in 10 minutes. Give an exact answer.

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -A bicycle with a 26-inch wheel (diameter) travels a distance of 300 feet. How many revolutions does the wheel make (to the nearest revolution) ?

Find the area of a sector of a circle having radius r and central angle $\theta$ . If necessary, express the answer to the nearest tenth. - $\mathrm { r } = 15.0 \mathrm {~m} , \theta = 20 ^ { \circ }$

Find the area of a sector of a circle having radius r and central angle $\theta$ . If necessary, express the answer to the nearest tenth. - $\mathrm { r } = 7.0 \mathrm { mi } , \theta = 268 ^ { \circ }$

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -A pulley with a diameter of 28 inches is driven by a belt which is moving $1035 \mathrm { ft } / \mathrm { min }$ . To the nearest unit, how many revolutions per minute are made by the pulley?

Find the area of a sector of a circle having radius r and central angle $\theta$ . If necessary, express the answer to the nearest tenth. - $\mathrm { r } = 4.0 \mathrm { ft } , \theta = \frac { 2 \pi } { 3 }$ radians

Approximate the length using the formula for arc length. Round to the nearest meter. -A television tower $420 \mathrm {~m}$ high subtends an angle of $4 ^ { \circ } 40 ^ { \prime }$ . How far away is the tower?

Find the area of a sector of a circle having radius r and central angle $\theta$ . If necessary, express the answer to the nearest tenth. - $\mathrm { r } = 47.2 \mathrm {~cm} , \theta = \frac { \pi } { 11 }$ radians

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -A pulley rotates through $50 ^ { \circ }$ in one minute. How many rotations (to the nearest tenth of a rotation) does the pulley make in an hour?

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -A car wheel has a 14 -inch radius. Through what angle (to the nearest tenth of a degree) does the wheel turn when the car rolls forward $3 \mathrm { ft }$ ?

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -Suppose the tip of the minute hand of a clock is 6 inches from the center of the clock. Determine the distance traveled by the tip of the minute hand in 30 minutes. Give an exact answer.

Approximate the length using the formula for arc length. Round to the nearest meter. -A tree $490 \mathrm {~m}$ away subtends an angle of $2 ^ { \circ }$ . Find the height of the tree.

Quiz 7: The Circular Functions and Their Graphs

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -Suppose the tip of the minute hand of a clock is 3 inches from the center of the clock. Determine the distance traveled by the tip of the minute hand in 4.54.54.5 hours. Give an exact answer.

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -Find the radius (to the nearest hundredth of a millimeter) of a pulley if rotating the pulley 74.53∘74.53 ^ { \circ }74.53∘ raises the pulley 16.0 mm16.0 \mathrm {~mm}16.0 mm .

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -The minute hand of a clock is 7 inches long. What distance does its tip move in 24 minutes? Give an exact answer.

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -Suppose the tip of the minute hand of a clock is 9 inches from the center of the clock. Determine the distance traveled by the tip of the minute hand in 10 minutes. Give an exact answer.

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -A bicycle with a 26-inch wheel (diameter) travels a distance of 300 feet. How many revolutions does the wheel make (to the nearest revolution) ?

Find the area of a sector of a circle having radius r and central angle θ\thetaθ . If necessary, express the answer to the nearest tenth. - r=15.0 m,θ=20∘\mathrm { r } = 15.0 \mathrm {~m} , \theta = 20 ^ { \circ }r=15.0 m,θ=20∘

Find the area of a sector of a circle having radius r and central angle θ\thetaθ . If necessary, express the answer to the nearest tenth. - r=7.0mi,θ=268∘\mathrm { r } = 7.0 \mathrm { mi } , \theta = 268 ^ { \circ }r=7.0mi,θ=268∘

Find the area of a sector of a circle having radius r and central angle θ\thetaθ . If necessary, express the answer to the nearest tenth. - r=4.0ft,θ=2π3\mathrm { r } = 4.0 \mathrm { ft } , \theta = \frac { 2 \pi } { 3 }r=4.0ft,θ=32π​ radians

Approximate the length using the formula for arc length. Round to the nearest meter. -A television tower 420 m420 \mathrm {~m}420 m high subtends an angle of 4∘40′4 ^ { \circ } 40 ^ { \prime }4∘40′ . How far away is the tower?

Find the area of a sector of a circle having radius r and central angle θ\thetaθ . If necessary, express the answer to the nearest tenth. - r=47.2 cm,θ=π11\mathrm { r } = 47.2 \mathrm {~cm} , \theta = \frac { \pi } { 11 }r=47.2 cm,θ=11π​ radians

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -A pulley rotates through 50∘50 ^ { \circ }50∘ in one minute. How many rotations (to the nearest tenth of a rotation) does the pulley make in an hour?

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -A car wheel has a 14 -inch radius. Through what angle (to the nearest tenth of a degree) does the wheel turn when the car rolls forward 3ft3 \mathrm { ft }3ft ?

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -Suppose the tip of the minute hand of a clock is 6 inches from the center of the clock. Determine the distance traveled by the tip of the minute hand in 30 minutes. Give an exact answer.

Approximate the length using the formula for arc length. Round to the nearest meter. -A tree 490 m490 \mathrm {~m}490 m away subtends an angle of 2∘2 ^ { \circ }2∘ . Find the height of the tree.

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -Suppose the tip of the minute hand of a clock is 3 inches from the center of the clock. Determine the distance traveled by the tip of the minute hand in $4.5$ hours. Give an exact answer.

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -Find the radius (to the nearest hundredth of a millimeter) of a pulley if rotating the pulley $74.53 ^ { \circ }$ raises the pulley $16.0 \mathrm {~mm}$ .

Find the area of a sector of a circle having radius r and central angle $\theta$ . If necessary, express the answer to the nearest tenth. - $\mathrm { r } = 15.0 \mathrm {~m} , \theta = 20 ^ { \circ }$

Find the area of a sector of a circle having radius r and central angle $\theta$ . If necessary, express the answer to the nearest tenth. - $\mathrm { r } = 7.0 \mathrm { mi } , \theta = 268 ^ { \circ }$

Find the area of a sector of a circle having radius r and central angle $\theta$ . If necessary, express the answer to the nearest tenth. - $\mathrm { r } = 4.0 \mathrm { ft } , \theta = \frac { 2 \pi } { 3 }$ radians

Approximate the length using the formula for arc length. Round to the nearest meter. -A television tower $420 \mathrm {~m}$ high subtends an angle of $4 ^ { \circ } 40 ^ { \prime }$ . How far away is the tower?

Find the area of a sector of a circle having radius r and central angle $\theta$ . If necessary, express the answer to the nearest tenth. - $\mathrm { r } = 47.2 \mathrm {~cm} , \theta = \frac { \pi } { 11 }$ radians

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -A pulley rotates through $50 ^ { \circ }$ in one minute. How many rotations (to the nearest tenth of a rotation) does the pulley make in an hour?

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -A car wheel has a 14 -inch radius. Through what angle (to the nearest tenth of a degree) does the wheel turn when the car rolls forward $3 \mathrm { ft }$ ?

Approximate the length using the formula for arc length. Round to the nearest meter. -A tree $490 \mathrm {~m}$ away subtends an angle of $2 ^ { \circ }$ . Find the height of the tree.