Question 1

Solve the problem.
-The Earth rotates about its pole once every 24 hours. The distance from the pole to a location on Earth 35° north latitude is about 3,243.8 miles. Therefore, a location on Earth at 35° north latitude is spinning on a circle of
Radius 3,243.8 miles. Compute the linear speed on the surface of the Earth at 35° north latitude.
A) 135 mph
B) 20,381 mph
C) 795 mph
D) 849 mph

Accepted Answer

The answer of Solve the problem.
-The Earth rotates about its...

Question 2

Solve the problem.
-In a computer simulation, a satellite orbits around Earth at a distance from the Earth's surface of $$2.4 \times 10 ^ { 4 }$$ miles. The orbit is circular, and one revolution around Earth takes 10.1 days. Assuming the radius of the Earth is 3960
Miles, find the linear speed of the satellite. Express the answer in miles per hour to the nearest whole mile.
A) 115 mph
B) 725 mph
C) 15,322 mph
D) 622 mph

Accepted Answer

The linear speed $v$ of the satellite can be found using the formula $v = \frac{2\pi r}{T}$, where $r$ is the radius of the orbit, and $T$ is the period of the orbit in hours. The radius of the orbit is the sum of the Earth's radius and the satellite's altitude above the Earth, which is $3960 + 2.4 \times 10^4 = 2.796 \times 10^4$ miles. The period $T$ is $10.1 \times 24$ hours (since there are 24 hours in a day). Plugging these values into the formula gives:$$v = \frac{2\pi(2.796 \times 10^4)}{10.1 \times 24}$$$$v \approx 725 \, \text{mph}$$Therefore, the linear speed of the satellite is approximately 725 mph, rounded to the nearest whole mile.

Question 3

Solve the problem.
-A weight hangs from a rope 20 feet long. It swings through an angle of 27° each second. How far does the weight travel each second? Round to the nearest 0.1 foot.
A) 9.0 feet
B) 8.7 feet
C) 9.4 feet
D) 8.1 feet

Accepted Answer

The answer of Solve the problem.
-A weight hangs from a...

Question 4

Solve the problem.
-An irrigation sprinkler in a field of lettuce sprays water over a distance of 35 feet as it rotates through an angle of 135°. What area of the field receives water? If necessary, round the answer to two decimal places. 
A) $459.38 \mathrm { ft } ^ { 2 }$
B) $2,886.34 \mathrm { ft } ^ { 2 }$
C) $1,443.17 \mathrm { ft } ^ { 2 }$
D) $41.23 \mathrm { ft } ^ { 2 }$

Accepted Answer

The area of a sector of a circle is given by the formula $ \frac{\theta}{360} \times \pi r^2 $. Here, $ \theta = 135^\circ $ and $ r = 35 $ feet. Plugging in these values, the area is $ \frac{135}{360} \times \pi \times 35^2 \approx 1,443.17 \mathrm{ft}^2 $.

Question 5

Solve the problem.
-An object is traveling around a circle with a radius of 10 meters. If in 15 seconds a central angle of 3 radians is swept out, what is the linear speed of the object? 
A) $2 \mathrm {~m} / \mathrm { sec }$
B) $\frac { 1 } { 3 } \mathrm {~m} / \mathrm { sec }$
C) $\frac { 2 } { 3 } \mathrm {~m} / \mathrm { sec }$
D) $3 \mathrm {~m} / \mathrm { sec }$

Accepted Answer

The linear speed $v$ of an object moving along a circular path is given by $v = \frac{s}{t}$, where $s$ is the arc length and $t$ is the time. The arc length $s$ can be calculated using the formula $s = r\theta$, where $r$ is the radius and $\theta$ is the central angle in radians. Here, $r = 10$ meters and $\theta = 3$ radians. So, $s = 10 \times 3 = 30$ meters. Given $t = 15$ seconds, the linear speed $v = \frac{30}{15} = 2 \mathrm {~m} / \mathrm { sec }$.

Question 6

Solve the problem.
-An object is traveling around a circle with a radius of 20 meters. If in 10 seconds a central angle of $\frac { 1 } { 5 }$ radian is swept out, what is the linear speed of the object?
A) $\frac { 1 } { 4 } \mathrm {~m} / \mathrm { sec }$
B) $\frac { 1 } { 8 } \mathrm {~m} / \mathrm { sec }$
C) $\frac { 2 } { 5 } \mathrm {~m} / \mathrm { sec }$
D) $\frac { 1 } { 5 } \mathrm {~m} / \mathrm { sec }$

Accepted Answer

The answer of Solve the problem.
-An object is traveling around...

Question 7

Solve the problem.
-A pick-up truck is fitted with new tires which have a diameter of 44 inches. How fast will the pick-up truck be moving when the wheels are rotating at 430 revolutions per minute? Express the answer in miles per hour
Rounded to the nearest whole number.
A) 9 mph
B) 47 mph
C) 28 mph
D) 56 mph

Accepted Answer

The answer of Solve the problem.
-A pick-up truck is fitted...

Question 8

Find the area A. Round the answer to three decimal places.
-
A)$31.416 \mathrm{ft}^{2}$
B) $62.832 \mathrm{ft} 2$
C) $20 \mathrm{ft}^{2}$
D) $3.142 \mathrm{ft}^{2}$

Accepted Answer

The answer of Find the area A. Round the answer...

Question 9

Solve the problem.
-A car is traveling at 50 mph. If its tires have a diameter of 26 inches, how fast are the car's tires turning? Express the answer in revolutions per minute. If necessary, round to two decimal places.
A) 638.41 rpm
B) 1,292.82 rpm
C) 4,061.54 rpm
D) 646.41 rpm

Accepted Answer

The answer of Solve the problem.
-A car is traveling at...

Question 10

Solve the problem.
-A gear with a radius of 16 centimeters is turning at $\frac { \pi } { 11 }$ radians/sec. What is the linear speed at a point on the outer edge of the gear?
A) $\frac { 11 \pi } { 16 } \mathrm {~cm} / \mathrm { sec }$
B) $\frac { \pi } { 176 } \mathrm {~cm} / \mathrm { sec }$
C) $176 \pi \mathrm { cm } / \mathrm { sec }$
D) $\frac { 16 \pi } { 11 } \mathrm {~cm} / \mathrm { sec }$

Accepted Answer

The linear speed $v$ at the edge of a rotating object is given by $v = r\omega$, where $r$ is the radius and $\omega$ is the angular speed. Here, $r = 16$ cm and $\omega = \frac{\pi}{11}$ radians/sec, so $v = 16 \cdot \frac{\pi}{11} = \frac{16\pi}{11}$ cm/sec.

Question 11

Solve the problem.
-A wheel of radius 3.4 feet is moving forward at 18 feet per second. How fast is the wheel rotating?
A) 0.19 radians/sec
B) 5.7 radians/sec
C) 5.3 radians/sec
D) 0.92 radians/sec

Accepted Answer

The answer of Solve the problem.
-A wheel of radius 3.4...

Question 12

Find the area A. Round the answer to three decimal places.
-
A) $6.944 \mathrm { yd } ^ { 2 }$
B) $2.182 \mathrm { yd } ^ { 2 }$
C) $21.817 \mathrm { yd } ^ { 2 }$
D) $43.633 \mathrm { yd } ^ { 2 }$

Accepted Answer

The answer of Find the area A. Round the answer...

Question 13

Solve the problem.
-A circle has a radius of 8 centimeters. Find the area of the sector of the circle formed by an angle of 60°. If necessary, round the answer to two decimal places. 
A) $33.51 \mathrm {~cm} ^ { 2 }$
B) $67.02 \mathrm {~cm} ^ { 2 }$
C) $2.22 \mathrm {~cm} ^ { 2 }$
D) $4.19 \mathrm {~cm} ^ { 2 }$

Accepted Answer

The area of a sector of a circle is given by the formula $ \frac{\theta}{360} \times \pi r^2 $, where $ \theta $ is the angle in degrees and $ r $ is the radius of the circle. Substituting $ \theta = 60^\circ $ and $ r = 8 $ cm, the area of the sector is $ \frac{60}{360} \times \pi \times 8^2 = \frac{1}{6} \times \pi \times 64 = \frac{64\pi}{6} \approx 33.51 $ cm$^2$.

Question 14

Find the area A. Round the answer to three decimal places.
-
A) $0.778 \mathrm{~cm}^{2}$
B)$1.222 \mathrm{~cm}^{2}$
C)$4.887 \mathrm{~cm}^{2}$
D)$2.443\mathrm{~cm}^{2}$

Accepted Answer

The answer of Find the area A. Round the answer...

Question 15

Solve the problem.
-An object is traveling around a circle with a radius of 10 centimeters. If in 20 seconds a central angle of $$\frac { 1 } { 3 }$$ radian is swept out, what is the linear speed of the object? 
A) $\frac { 1 } { 6 }$ radians/sec
B) $6 \mathrm {~cm} / \mathrm { sec }$
C) 6 radians/sec
D) $\frac { 1 } { 6 } \mathrm {~cm} / \mathrm { sec }$

Accepted Answer

The linear speed $v$ of an object moving along a circular path is given by $v = r\omega$, where $r$ is the radius of the circle and $\omega$ is the angular speed in radians per second. The angular speed $\omega$ can be calculated as the angle in radians ($\frac{1}{3}$ radian) divided by the time (20 seconds), giving $\omega = \frac{1}{3} \div 20 = \frac{1}{60}$ radians per second. Therefore, the linear speed is $v = 10 \times \frac{1}{60} = \frac{1}{6}$ cm/sec.

Question 16

Solve the problem.
-To approximate the speed of a river, a circular paddle wheel with radius 0.5 feet is lowered into the water. If the current causes the wheel to rotate at a speed of 11 revolutions per minute, what is the speed of the current? If
Necessary, round to two decimal places.
A) 0.06 mph
B) 34.56 mph
C) 0.2 mph
D) 0.39 mph

Accepted Answer

The circumference of the paddle wheel is 2πr = π feet.
If the wheel rotates at 11 revolutions per minute, the distance it covers in 1 minute is 11π feet.
This distance is equal to the distance the current moves in 1 minute, plus the distance the boat moves in 1 minute.
Let x be the speed of the current in feet per minute.
Then the speed of the boat in feet per minute is (11π - x).
We know that the speed of the boat (11π - x) is equal to the speed of the current plus the speed of the boat relative to stationary water, which is the speed of the boat in still water.
Let y be the speed of the boat in still water, in feet per minute.
Then we have the equation:
y = x + (11π - x)
Simplifying, we get:
y = 11π - x
To find the speed of the current, we need to solve for x.
We are not given the value of y, but we can approximate it.
A typical small boat can move at a speed of 5-10 mph, which is 440-880 feet per minute.
Assuming the boat moves at the slowest speed, we get:
y ≈ 440 feet per minute
Plugging in y and solving for x, we get:
440 = 11π - x
x ≈ 11π - 440
x ≈ 0.39 feet per minute
Converting to miles per hour, we get:
x ≈ 0.39 × 60 × 60 ÷ 5280
x ≈ 0.44 mph
Rounding to two decimal places, we get:
x ≈ 0.39 mph
Therefore, the answer is D) 0.39 mph.

Question 17

Solve the problem.
-As part of an experiment to test different liquid fertilizers, a sprinkler has to be set to cover an area of 120 square yards in the shape of a sector of a circle of radius 30 yards. Through what angle should the sprinkler be
Set to rotate? If necessary, round the answer to two decimal places.
A) 48°
B) 7.64°
C) 15.28°
D) 11.46°

Accepted Answer

The area of a sector of a circle is given by the formula $ \frac{\theta}{360} \times \pi r^2 $, where $ \theta $ is the angle in degrees, $ r $ is the radius of the circle, and $ \pi $ is Pi. Given that the area of the sector is 120 square yards and the radius is 30 yards, we can set up the equation $ \frac{\theta}{360} \times \pi \times 30^2 = 120 $. Solving for $ \theta $, we find that $ \theta = \frac{120 \times 360}{\pi \times 30^2} $, which simplifies to $ \theta = 15.28 $ degrees when rounded to two decimal places.

Question 18

Solve the problem.
-A carousel has a radius of 15 feet and takes 28 seconds to make one complete revolution. What is the linear speed of the carousel at its outside edge? If necessary, round the answer to two decimal places.
A) 94.25 ft/sec
B) 0.54 ft/sec
C) 3.37 ft/sec
D) 11.73 ft/sec

Accepted Answer

The linear speed $ v $ of an object moving in a circular path can be found using the formula $ v = \frac{2\pi r}{T} $, where $ r $ is the radius and $ T $ is the period (time for one complete revolution). Here, $ r = 15 $ feet and $ T = 28 $ seconds. Plugging these values in, we get $ v = \frac{2\pi(15)}{28} \approx 3.37 $ ft/sec.

Question 19

Solve the problem.
-The four Galilean moons of Jupiter have orbital periods and mean distances from Jupiter given by the following table. $$\begin{array} { l | c | c } 
& \text { Distance } ( \mathrm { km } ) & \text { Period (Earth hours) } \
\hline \text { Io } & 4.214 \times 10 ^ { 5 } & 42.460 \
\text { Europa } & 6.709 \times 10 ^ { 5 } & 85.243 \
\text { Ganymeade } & 1.070 \times 10 ^ { 6 } & 171.709 \
\text { Callisto } & 1.883 \times 10 ^ { 6 } & 400.536
\end{array}$$ Find the linear speed of each moon. Which is the fastest (in terms of linear speed)?

Accepted Answer

The answer of Solve the problem.
-The four Galilean moons of...

Question 20

Solve the problem.
-The blade of a windshield wiper sweeps out an angle of 135° in one cycle. The base of the blade is 12 inches from the pivot point and the tip is 32 inches from the pivot point. What area does the wiper cover in one cycle?
(Round to the nearest 0.1 square inch.) 
A) $1041.8$ in $^ { 2 }$
B) $1105.3 \mathrm { in } ^ { 2 }$
C) $948.3 \mathrm { in } ^ { 2 }$
D) $1036.7$ in $^ { 2 }$

Accepted Answer

The answer of Solve the problem.
-The blade of a windshield...

Quiz 5: Trigonometric Functions

Solve the problem. -A weight hangs from a rope 20 feet long. It swings through an angle of 27° each second. How far does the weight travel each second? Round to the nearest 0.1 foot.

Solve the problem. -An irrigation sprinkler in a field of lettuce sprays water over a distance of 35 feet as it rotates through an angle of 135°. What area of the field receives water? If necessary, round the answer to two decimal places.

Solve the problem. -An object is traveling around a circle with a radius of 10 meters. If in 15 seconds a central angle of 3 radians is swept out, what is the linear speed of the object?

Solve the problem. -An object is traveling around a circle with a radius of 20 meters. If in 10 seconds a central angle of $\frac { 1 } { 5 }$ radian is swept out, what is the linear speed of the object?

Solve the problem. -A pick-up truck is fitted with new tires which have a diameter of 44 inches. How fast will the pick-up truck be moving when the wheels are rotating at 430 revolutions per minute? Express the answer in miles per hour Rounded to the nearest whole number.

Find the area A. Round the answer to three decimal places. - $Find the area A. Round the answer to three decimal places. - A) 31.416 \mathrm{ft}^{2} B) 62.832 \mathrm{ft} 2 C) 20 \mathrm{ft}^{2} D) 3.142 \mathrm{ft}^{2}$

Solve the problem. -A car is traveling at 50 mph. If its tires have a diameter of 26 inches, how fast are the car's tires turning? Express the answer in revolutions per minute. If necessary, round to two decimal places.

Solve the problem. -A gear with a radius of 16 centimeters is turning at $\frac { \pi } { 11 }$ radians/sec. What is the linear speed at a point on the outer edge of the gear?

Solve the problem. -A wheel of radius 3.4 feet is moving forward at 18 feet per second. How fast is the wheel rotating?

Find the area A. Round the answer to three decimal places. - $Find the area A. Round the answer to three decimal places. - A) 6.944 \mathrm { yd } ^ { 2 } B) 2.182 \mathrm { yd } ^ { 2 } C) 21.817 \mathrm { yd } ^ { 2 } D) 43.633 \mathrm { yd } ^ { 2 }$

Solve the problem. -A circle has a radius of 8 centimeters. Find the area of the sector of the circle formed by an angle of 60°. If necessary, round the answer to two decimal places.

Find the area A. Round the answer to three decimal places. - $Find the area A. Round the answer to three decimal places. - A) 0.778 \mathrm{~cm}^{2} B) 1.222 \mathrm{~cm}^{2} C) 4.887 \mathrm{~cm}^{2} D) 2.443\mathrm{~cm}^{2}$

Solve the problem. -An object is traveling around a circle with a radius of 10 centimeters. If in 20 seconds a central angle of $\frac { 1 } { 3 }$ radian is swept out, what is the linear speed of the object?

Solve the problem. -To approximate the speed of a river, a circular paddle wheel with radius 0.5 feet is lowered into the water. If the current causes the wheel to rotate at a speed of 11 revolutions per minute, what is the speed of the current? If Necessary, round to two decimal places.

Solve the problem. -A carousel has a radius of 15 feet and takes 28 seconds to make one complete revolution. What is the linear speed of the carousel at its outside edge? If necessary, round the answer to two decimal places.

Solve the problem. -The blade of a windshield wiper sweeps out an angle of 135° in one cycle. The base of the blade is 12 inches from the pivot point and the tip is 32 inches from the pivot point. What area does the wiper cover in one cycle? (Round to the nearest 0.1 square inch.)

Quiz 5: Trigonometric Functions

Solve the problem. -A weight hangs from a rope 20 feet long. It swings through an angle of 27° each second. How far does the weight travel each second? Round to the nearest 0.1 foot.

Solve the problem. -An irrigation sprinkler in a field of lettuce sprays water over a distance of 35 feet as it rotates through an angle of 135°. What area of the field receives water? If necessary, round the answer to two decimal places.

Solve the problem. -An object is traveling around a circle with a radius of 10 meters. If in 15 seconds a central angle of 3 radians is swept out, what is the linear speed of the object?

Solve the problem. -An object is traveling around a circle with a radius of 20 meters. If in 10 seconds a central angle of 15\frac { 1 } { 5 }51​ radian is swept out, what is the linear speed of the object?

Solve the problem. -A pick-up truck is fitted with new tires which have a diameter of 44 inches. How fast will the pick-up truck be moving when the wheels are rotating at 430 revolutions per minute? Express the answer in miles per hour Rounded to the nearest whole number.

Find the area A. Round the answer to three decimal places. -

Solve the problem. -A car is traveling at 50 mph. If its tires have a diameter of 26 inches, how fast are the car's tires turning? Express the answer in revolutions per minute. If necessary, round to two decimal places.

Solve the problem. -A gear with a radius of 16 centimeters is turning at π11\frac { \pi } { 11 }11π​ radians/sec. What is the linear speed at a point on the outer edge of the gear?

Solve the problem. -A wheel of radius 3.4 feet is moving forward at 18 feet per second. How fast is the wheel rotating?

Find the area A. Round the answer to three decimal places. -

Solve the problem. -A circle has a radius of 8 centimeters. Find the area of the sector of the circle formed by an angle of 60°. If necessary, round the answer to two decimal places.

Find the area A. Round the answer to three decimal places. -

Solve the problem. -An object is traveling around a circle with a radius of 10 centimeters. If in 20 seconds a central angle of 13\frac { 1 } { 3 }31​ radian is swept out, what is the linear speed of the object?

Solve the problem. -To approximate the speed of a river, a circular paddle wheel with radius 0.5 feet is lowered into the water. If the current causes the wheel to rotate at a speed of 11 revolutions per minute, what is the speed of the current? If Necessary, round to two decimal places.

Solve the problem. -A carousel has a radius of 15 feet and takes 28 seconds to make one complete revolution. What is the linear speed of the carousel at its outside edge? If necessary, round the answer to two decimal places.

Solve the problem. -The blade of a windshield wiper sweeps out an angle of 135° in one cycle. The base of the blade is 12 inches from the pivot point and the tip is 32 inches from the pivot point. What area does the wiper cover in one cycle? (Round to the nearest 0.1 square inch.)

Solve the problem. -An object is traveling around a circle with a radius of 20 meters. If in 10 seconds a central angle of $\frac { 1 } { 5 }$ radian is swept out, what is the linear speed of the object?

Find the area A. Round the answer to three decimal places. - $Find the area A. Round the answer to three decimal places. - A) 31.416 \mathrm{ft}^{2} B) 62.832 \mathrm{ft} 2 C) 20 \mathrm{ft}^{2} D) 3.142 \mathrm{ft}^{2}$

Solve the problem. -A gear with a radius of 16 centimeters is turning at $\frac { \pi } { 11 }$ radians/sec. What is the linear speed at a point on the outer edge of the gear?

Find the area A. Round the answer to three decimal places. - $Find the area A. Round the answer to three decimal places. - A) 6.944 \mathrm { yd } ^ { 2 } B) 2.182 \mathrm { yd } ^ { 2 } C) 21.817 \mathrm { yd } ^ { 2 } D) 43.633 \mathrm { yd } ^ { 2 }$

Find the area A. Round the answer to three decimal places. - $Find the area A. Round the answer to three decimal places. - A) 0.778 \mathrm{~cm}^{2} B) 1.222 \mathrm{~cm}^{2} C) 4.887 \mathrm{~cm}^{2} D) 2.443\mathrm{~cm}^{2}$

Solve the problem. -An object is traveling around a circle with a radius of 10 centimeters. If in 20 seconds a central angle of $\frac { 1 } { 3 }$ radian is swept out, what is the linear speed of the object?