Q: A child spins a rock at the end of a $2$-ft rope at a rate of $180$ rpm. What is the linear speed of the rock when it is released? (Use $v = r\omega$, $\omega$ in rad/min)

$2 \times 2\pi \times 180 = 2,261.95$ ft/min

Q: A car tire with a radius of $0.3$ m rotates at $120$ rpm. What is the linear speed of a point on the edge of the tire? (Use $v = r\omega$ and $\omega$ in rad/min)

$0.3 \times 2\pi \times 120 = 226$ m/min

Question 1

Which of the following best defines a central angle in a circle?

Accepted Answer

An angle whose vertex is at the center of the circle.

Question 2

Convert $150^{\circ}$ to radians.

Accepted Answer

$\frac{5\pi}{6}$

Question 3

What is the formula for the arc length $s$ of a circle with radius $r$ intercepted by a central angle $	heta$ in radians?

Accepted Answer

$s = r	heta$

Question 4

A circle has a radius of $3$ m and a central angle of $100^{\circ}$. What is the arc length intercepted by this angle? (Use $\pi \approx 3.14$)

Accepted Answer

$5.24$ m

Question 5

Which of the following is the correct conversion factor to change degrees to radians?

Accepted Answer

$\frac{\pi}{180}$

Question 6

Find the area of a sector of a circle with radius $6$ inches and central angle $45^{\circ}$.

Accepted Answer

$14.14$ in$^2$

Question 7

If a wheel rotates $270^{\circ}$ counterclockwise, what is the equivalent angle in radians?

Accepted Answer

$\frac{3\pi}{2}$

Question 8

A child spins a rock at the end of a $2$-ft rope at a rate of $180$ rpm. What is the linear speed of the rock when it is released? (Use $v = r\omega$, $\omega$ in rad/min)

Accepted Answer

$2 	imes 2\pi 	imes 180 = 2,261.95$ ft/min

Question 9

What is the angular speed in radians per second of an object rotating at $900$ rpm?

Accepted Answer

$94.25$ rad/s

Question 10

Which formula gives the area $A$ of a sector of a circle with radius $r$ and central angle $	heta$ in radians?

Accepted Answer

$A = \frac{1}{2} r^2 	heta$

Question 11

Convert $\frac{\pi}{4}$ radians to degrees.

Accepted Answer

$45^{\circ}$

Question 12

A car tire with a radius of $0.3$ m rotates at $120$ rpm. What is the linear speed of a point on the edge of the tire? (Use $v = r\omega$ and $\omega$ in rad/min)

Accepted Answer

$0.3 	imes 2\pi 	imes 120 = 226$ m/min

Question 13

If the arc length of a circle is $10$ cm and the radius is $4$ cm, what is the measure of the central angle in radians?

Accepted Answer

$2.5$ radians

Question 14

Which of the following statements is true about the relationship between arc length and central angle?

Accepted Answer

Arc length is directly proportional to the central angle in radians.

Question 15

Convert $6.5$ yards to centimeters. (1 in = 2.54 cm, 1 yd = 36 in)

Accepted Answer

$594.4$ cm

Angles, Radian Measure, Arc Length, and Speed in Circles

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