1982 AMC 12 Problem 22

Question

In a narrow alley of width

w

a ladder of length

a

is placed with its foot at a point

P

between the walls. Resting against one wall at

Q,

a distance

k

above the ground, the ladder makes a

45^\circ

angle with the ground. Resting against the other wall at

R,

a distance

h

above the ground, the ladder makes a

75^\circ

angle with the ground. The width

w

is equal to

a

RQ

k

\frac{h+k}{2}

h

Express the two horizontal portions of the alley using cosines

Compare

\cos75^\circ+\cos45^\circ

with

\sin75^\circ

The horizontal distances from

P

to the walls are

a\cos75^\circ

and

a\cos45^\circ,

so

w=a(\cos75^\circ+\cos45^\circ).

The sum-to-product identity gives

\cos75^\circ+\cos45^\circ=\sin75^\circ.

Since

h=a\sin75^\circ,

w=h.

Therefore, the correct answer is E.

Problem 22 in other years