Question

A carnival ride is in the shape of a wheel with a radius of 15 feet. The wheel has 24 cars attached to the center of the wheel. What is the central angle, arc length, and area of a sector between any two cars? Round answers to the nearest hundredth if applicable. You must show all work and calculations to receive credit.

Answer 1

Each central angle will have measure
`(2\pi)/(24)=\frac\pi{12}\text{ rad}=15^\circ`.

The arc length
`L` of each 1/24-th section of the wheel occurs in the following ratio with the whole wheel's circumference:


`\frac{2\pi\text{ rad}}{2\pi(15)\text{ ft}}=\frac{\frac\pi{12}\text{ rad}}L\implies\frac1{15}=\frac\pi{12L}\implies L=(15\pi)/(12)\text{ ft}`

`L\approx3.93\text{ ft}`

The area of the sector
`A` occurs in a similar ratio with the wheel's total area:


`\frac{2\pi\text{ rad}}{\pi(15)^2\text{ ft}^2}=\frac{\frac\pi{12}\text{ rad}}A\implies\frac2{225}=\frac\pi{12A}\implies A=\frac{75\pi}8\text{ ft}^2`

`A\approx29.45\text{ ft}^2`