LazyLimabean
contestada

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.

Respuesta :

LammettHash
Each central angle will have measure [tex]\dfrac{2\pi}{24}=\dfrac\pi{12}\text{ rad}=15^\circ[/tex].

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

[tex]\dfrac{2\pi\text{ rad}}{2\pi(15)\text{ ft}}=\dfrac{\frac\pi{12}\text{ rad}}L\implies\dfrac1{15}=\dfrac\pi{12L}\implies L=\dfrac{15\pi}{12}\text{ ft}[/tex]
[tex]L\approx3.93\text{ ft}[/tex]

The area of the sector [tex]A[/tex] occurs in a similar ratio with the wheel's total area:

[tex]\dfrac{2\pi\text{ rad}}{\pi(15)^2\text{ ft}^2}=\dfrac{\frac\pi{12}\text{ rad}}A\implies\dfrac2{225}=\dfrac\pi{12A}\implies A=\dfrac{75\pi}8\text{ ft}^2[/tex]
[tex]A\approx29.45\text{ ft}^2[/tex]