Question

a boat is heading towards a lighthouse, where charlotte is watching from a vertical distance of 137 feet above the water. charlotte measures an angle of depression to the boat at point a to be 11°. at some later time, charlotte takes another measurement and finds the angle of depression to the boat (now at point b) to be 47°. find the distance from point a to point b. round your answer to the nearest foot if necessary. answer attempt 2 out of 2

Explanation:

Step1: Understand the angle of depression

The angle of depression is equal to the angle of elevation from the boat to Charlotte. Let the height of the lighthouse (Charlotte's vertical distance above water) be \( h = 137 \) feet. Let the horizontal distance from the base of the lighthouse to point \( A \) be \( x_A \) and to point \( B \) be \( x_B \).
Using the tangent function for right triangles: \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \), so for angle of elevation \( \theta \), \( \tan(\theta)=\frac{h}{x} \), which gives \( x = \frac{h}{\tan(\theta)} \).

Step2: Calculate \( x_A \) (distance from lighthouse to A)

For point \( A \), the angle of elevation \( \theta_A = 11^\circ \). So \( x_A=\frac{137}{\tan(11^\circ)} \).
\( \tan(11^\circ)\approx0.1944 \), so \( x_A=\frac{137}{0.1944}\approx704.73 \) feet.

Step3: Calculate \( x_B \) (distance from lighthouse to B)

For point \( B \), the angle of elevation \( \theta_B = 47^\circ \). So \( x_B=\frac{137}{\tan(47^\circ)} \).
\( \tan(47^\circ)\approx1.0724 \), so \( x_B=\frac{137}{1.0724}\approx127.74 \) feet.

Step4: Calculate distance from A to B

Since the boat is moving towards the lighthouse, the distance \( AB = x_A - x_B \).
\( AB = 704.73 - 127.74 = 576.99\approx577 \) feet.

Answer:

\( 577 \)