Question

1. indirect measurement kenneth is sitting at the end of a pier and using binoculars to watch a whale surface. the pier is 30 feet above the water, and kenneth’s eye level is 3 feet above the pier. if the angle of depression to the whale is 20°, how far is the whale from kenneth’s binoculars? round your answer to the nearest tenth of a foot.

Explanation:

Step1: Identify total vertical height

First, calculate the total vertical distance from Kenneth's eye level to the water surface.
$30 + 3 = 33$ feet

Step2: Relate to angle of depression

The angle of depression equals the angle of elevation from the whale to Kenneth's eyes, so we use the sine function, where $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$. Here, the opposite side is the total vertical height, and the hypotenuse is the distance we need to find (let this distance be $d$).
$\sin(20^\circ) = \frac{33}{d}$

Step3: Solve for distance $d$

Rearrange the formula to isolate $d$.
$d = \frac{33}{\sin(20^\circ)}$
Calculate $\sin(20^\circ) \approx 0.3420$, then substitute:
$d \approx \frac{33}{0.3420} \approx 96.5$

Answer:

96.5 feet