Answer:
The height of the cliff is approximately 915 feet.
Step-by-step explanation:
Refer to the diagram attached. Let B represent the point on the ground where the cliff measures an angle of elevation of 56°. Let AH be the height of triangle ABC on the base BC. H is on both line AH and line BC.
The three angles of triangle ABC will be:
- [tex]\rm \hat{A} = 18^{\circ}[/tex];
- [tex]\rm A\hat{B}C = 180^{\circ} - 56^{\circ}[/tex];
- [tex]\rm \hat{C} = 18^{\circ}[/tex].
Only the length of segment BC is known. To find the height of the cliff, start by finding the length of segment AB. Apply the law of sine.
[tex]\displaystyle \rm AB = BC\times\frac{\sin{\hat{C}}}{\sin{\hat{A}}} = 2,200\times\frac{\sin{18^{\circ}}}{\sin{38^{\circ}}} \approx 1.10\times 10^{3} \; ft[/tex].
In the triangle ABH,
- AB is the hypotenuse, and
- AH is the side opposite to the angle [tex]\rm A\hat{B}H[/tex].
[tex]\rm AH = AB\times \sin{A\hat{B}H} = 1.10\times 10^{3} \times \sin{56^{\circ}} \approx 915\; ft[/tex].
