Answer Key

  1. A pendulum of length 13 in. swings through an angle of 24°. How far does the tip of the pendulum travel?

\[s= \frac{24^\circ}{360^\circ}\cdot2\pi(13\text{ in}) = 5.4 \text{ in}\]

  1. A wedged-shaped piece is cut from a pizza l0 in. in diameter so that the rounded edge of the crust measures 4.4 in. Determine the measure to the nearest degree of the angle at the pointed end of the piece of pizza.

\[4.4 \text{ in}= \frac{\theta}{360^\circ}\cdot2\pi(5\text{ in})\]

\[\theta=\frac{4.4 \text{ in}}{2\pi(5\text{ in})}\cdot360^\circ = 50.42 ^\circ\]

  1. The minute hand on a clock is 11.5 cm long. How far does its tip move between 4:00 and 4:15?

\[s=\theta r = \frac{\pi}{2}(11.5\text{ cm})=18.06 \text{ cm}\]

  1. Find the area of the sector of a circle of radius 2.2 cm that is intercepted by a central angle of 14°.

\[k=\frac{14^\circ}{360^\circ}\pi (2.2 \text{ cm})^2 = 0.59 \text{ cm}^2\]

  1. The area of a circle is 36 \(\pi\). Determine to the nearest square meter the area of the sector intercepted by a central angle of 50°.

\[k=\frac{50^\circ}{360^\circ}(36 \pi \text{ m}^2)= 5 \pi \text{ m}^2\]

  1. The wheel of a car has a 15-in. radius. Through what angle (in radians) does a point on the wheel rotate as the car travels 1 mile?

\[1 \text{ mi}=\theta(15\text{ in})\] \[\theta = \frac{1 \text{ mi}}{15\text{ in}}= \frac{1 \text{ mi}}{15\text{ in}}\cdot\frac{12\text{ in}}{1\text{ ft}}\cdot\frac{5280\text{ ft}}{1\text{ mi}} \approx 4224 \text{ radians}\]

  1. Determine the angular velocity in radians per second of a wheel turning at 78.6 revolutions per minute.

\[av=\frac{78.6 \text{ rot}}{1 \text{ min}}\cdot\frac{2\pi \text{ radians}}{1 \text{ rot}}\cdot\frac{1 \text{ min}}{60 \text{ sec}}\approx 8.23 \text{ rad/sec}\]

  1. Determine the number of revolutions per minute of a wheel with an angular velocity of 57 rad/s.

\[\text{rpm} = \frac{57 \text{ rad}}{1 \text{ sec}}\cdot \frac{1 \text{ rot}}{2\pi \text{ rad}}\cdot \frac{60 \text{ sec}}{1 \text{ min}} \approx 544.31 \text{ rpm}\]

  1. Determine the number of revolutions per minute of a wheel with an angular velocity of 300 deg/s.

\[\frac{300^\circ}{1 \text{ sec}}\cdot\frac{1 \text{ rot}}{360^\circ}\cdot\frac{60 \text{ sec}}{1 \text{ min}} = 50 \text{ rpm}\]

  1. If an engine is turning at 1000 rpm, what is the angular velocity of the engine’s crankshaft in radians per second?

\[\frac{1000 \text{ rot}}{1 \text{ min}}\cdot\frac{2\pi \text{ rad}}{1\text{ rot}}\cdot\frac{1 \text{ min}}{60 \text{ sec}} \approx 104.7 \text{ rad/sec}\]