Question

David wants to measure the width of a road

he chooses two points, A and B, 40m apart.
The angles made with a lamppost, P, on the opposite side are measured.
Angle BAP=45 degrees
Angle ABP=60 degrees
Work out the width, in metres, of the road. give your answer correct to 3 significant figures

Answer 1

The width of the road = 25.359 m

Explanation:

See the attached figure which represents the problem.

The length of AB = 40 m

let the width of the road w.

Construct PC perpendicular to AB, So, the measure of angle C = 90°

∠B = 60° and ∠A = 45°

Let the length of AC = x, so, the length of BC = 40-x

At ΔBCP which is a right triangle at C

tan B = opposite/adjacent = w/(40-x)

w = (40-x) * tan B ⇒(1)

At ΔACP which is a right triangle at C

tan A = opposite/adjacent = w/x

w = x * tan A ⇒(2)

from (1) and (2)

(40-x) * tan B = x * tan A

40 tan B - x *tan B = x tan A

40 tan B = x tan A + x *tan B

40 tan B = x (tan A + tan B)

x = (40 tan B)/(tan A + tan B) = (40 tan 60)/(tan 60 + tan 45) = 25.359 m

substitute at (2) with x

w = x tan A = 25.359 tan 45 = 25.359 m

So, The width of the road = 25.359 m