2002 AMC 10B Problem 24

Question

Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius

20

feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point

10

vertical feet above the bottom?

5

6

7.5

10

15

Answer: D

Concepts:circle special right triangle ratio and proportion

Difficulty rating: 1580

Small hint:

Place the center at height

20

; the rider rises

10

feet from the bottom

Big hint:

The rider is now

10

feet below center; find the central angle turned from the bottom

Solution:

Put the center

O

at height

20.

The bottom

A

is at height

0,

and the rider reaches height

10,

which is

10

feet below the center.

Draw a horizontal segment from the rider to the wheel’s vertical diameter. The resulting right triangle has a vertical leg of length

10

and a hypotenuse (the radius) of length

20.

That leg is half the hypotenuse, so the radius to the rider makes

60^\circ

with the downward vertical.

The wheel turns

360^\circ

in

60

seconds, so turning

60^\circ

takes

\dfrac{60}{360}\cdot 60 = 10

seconds.

Thus, the correct answer is D.