Question

question
a boat is heading towards a lighthouse, where charlotte is watching from a vertical distance of 127 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.

Explanation:

Step1: Recall angle of depression

The angle of depression is equal to the angle of elevation from the boat to Charlotte. Let the vertical height (from water to Charlotte) be \( h = 127 \) feet. Let the horizontal distance from the boat to the base of the lighthouse at point A be \( x_A \), and at point B be \( x_B \). Using the tangent function (\( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \)), for angle of elevation \( \theta \), \( \tan\theta=\frac{h}{x} \), so \( x = \frac{h}{\tan\theta} \).

Step2: Calculate \( x_A \) (distance at point A)

For \( \theta = 11^\circ \), \( x_A=\frac{127}{\tan(11^\circ)} \). Calculate \( \tan(11^\circ)\approx0.1944 \), so \( x_A\approx\frac{127}{0.1944}\approx653.3 \) feet.

Step3: Calculate \( x_B \) (distance at point B)

For \( \theta = 47^\circ \), \( x_B=\frac{127}{\tan(47^\circ)} \). Calculate \( \tan(47^\circ)\approx1.0724 \), so \( x_B\approx\frac{127}{1.0724}\approx118.4 \) feet.

Step4: Find distance between A and B

The distance \( AB = x_A - x_B \) (since the boat is moving towards the lighthouse, \( x_A > x_B \)). So \( AB\approx653.3 - 118.4 = 534.9 \approx 535 \) feet.

Answer:

\( 535 \)