1966 AMC 12 Problem 34

Question

Let

r

be the speed in miles per hour at which a wheel,

11

feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by

\dfrac14

of a second, the speed

r

is increased by

5

miles per hour. Then

r

is:

9

10

10\dfrac12

11

12

At

r

miles per hour, one rotation takes

\frac{7.5}{r}

seconds

Set

\frac{7.5}{r}-\frac{7.5}{r+5}=\frac{1}{4}

Since

11

feet is

\frac{11}{5280}

mile, one rotation at

r

mph takes

\frac{3600\cdot11}{5280r}=\frac{7.5}{r}

seconds. Thus

\frac{7.5}{r}-\frac{7.5}{r+5}=\frac14.

This reduces to

r^2+5r-150=0,

or

(r-10)(r+15)=0.

The positive speed is

r=10.

Therefore, the correct answer is B.

Problem 34 in other years