Question

The angle of elevation from a point on the ground to the top of a pyramid is 31 degrees 10​' The angle of elevation from a point 120 feet farther back to the top of the pyramid is 24 degrees 10​'. Find the height of the pyramid.

Answer 1

h ≈ 209 ft

Explanation:

The illustration forms 2 right angle triangle with same height , which is the height of the pyramid.

The first triangle has an unknown adjacent side and height but the angle is 31° 10'. The larger triangle has adjacent side of 120 ft plus the adjacent side of the smaller triangle.

let

the adjacent side of the smaller triangle = a

recall both triangle have same height = h

Therefore,

tan 31° 10' = opposite/adjacent = h/a

tan 31.167 = h/a

h = a tan 31.167

h = 0.6048345941 a

The larger triangle

adjacent side = 120 + a

height = h

tan 24° 10' = h/120 + a

convert the minutes to degree

tan 24.167° = h/120 + a

cross multiply

120 + a tan 24.167 = h

120 + a (0.44872570999 ) = h

h = 53.8470851998 + 0.44872570999a

Therefore,

0.6048345941 a = 53.8470851998 + 0.44872570999a

0.6048345941 a - 0.44872570999a = 53.8470851998

0.1561094942 a = 53.8470851998

divide both sides by 0.1561094942

a = 53.8470851998 / 0.1561094942

a = 344.931520506

a ≈ 345 ft

The height can be computed as

tan 31.167 = h/a

h = 345 × tan 31.167

h = 345 × 0.6048345941

h = 208.667934967

h ≈ 209 ft