Question

From a ship, the angle of elevation of a point, A, at the top of a cliff is 21°. After the ship has sailed 2,500 feet directly toward the foot of the cliff, the angle of elevation of ∠A is 47°. (Assume the cliff is perpendicular to the ground.) The height of the cliff is __ feet.

Answer 1

Answer:

10701.97 feet.

Step-by-step explanation:

Refer the attached figure

The angle of elevation of a point, A, at the top of a cliff is 21° i.e. ∠ACB = 21°

The ship has sailed 2,500 feet directly toward the foot of the cliff i.e. CD = 2500 feet.

Then the angle of elevation becomes 47° i.e.∠ADB = 47°

Let BD be x

So, BC = BD+DC=x+2500

Let the height of the cliff be h feet.

In ΔABD

We will use trigonometric ratios

$tan\theta = \frac{Perpendicular}{Base}$

$tan47^{\circ} = \frac{AB}{BD}$

$1.072= \frac{h}{x}$

$1.072x=h$    ---a

In ΔABC

We will use trigonometric ratios

$tan\theta = \frac{Perpendicular}{Base}$

$tan21^{\circ} = \frac{AB}{BC}$

$0.869 = \frac{h}{x+2500}$

$0.869(x+2500) =h$

$0.869x+2172.5 =h$   -----b

Equate a and b

$0.869x+2172.5 =1.072x$

$2172.5 =1.072x-0.869x$

$2172.5 =0.203x$

$\frac{2172.5}{0.203}=x$

$10701.97=x$

Thus the height of the cliff is 10701.97 feet.