If the height of the tower is H meters find an equation for k the horizontal distance between point B and the base of the castle

Question

A safety fence is constructed to protect tourist from the danger of an eroding castle toppling down at point a the surveyor takes an angle measurement to the top of the tower of 14° then she walks 28 m towards the tower and takes another reading of 25° at point B

Answer 1

Using a diagram with your labels,

tan 14° = H / (28 + k)

(28 + k) tan 14° = H

28 tan 14° + k tan 14° = H

tan 25° = H / k

k tan 25° = H

k tan 25° = 28 tan 14° + k tan 14°

k tan 25° - k tan 14° = 28 tan 14°

k (tan 25° - tan 14°) = 28 tan 14°

k = (28 tan 14°) / (tan 25° - tan 14°)

Answer 2

The distance from point a to be is 28 m, so the distance from a to the tower is 28 + k. The height h in terms of the 14° angel is H = (k + 28) tan14.

H can also be expressed as H = k tan 25°.

Equating both expressions for H gives k tan 25 = (k+ 28) tan 14 which can be solved for k.

k = 28 tan14 / (tan 25 - tan 14) = 32 m