Question

a boat is heading towards a lighthouse, where yaritza is watching from a vertical distance of 139 feet above the water. yaritza measures an angle of depression to the boat at point a to be 21°. at some later time, yaritza takes another measurement and finds the angle of depression to the boat (now at point b) to be 50°. find the distance from point a to point b. round your answer to the nearest foot if necessary.

Explanation:

Step1: Understand the problem with trigonometry

We have a right triangle situation where the vertical height (from Yaritza to water) is 139 feet. The angle of depression is equal to the angle of elevation from the boat to Yaritza (alternate interior angles). Let's denote the horizontal distance from the lighthouse to point A as \( x_A \) and to point B as \( x_B \). We can use the tangent function: \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}=\frac{139}{x} \), so \( x = \frac{139}{\tan(\theta)} \).

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

For point A, \( \theta = 21^\circ \). So \( x_A=\frac{139}{\tan(21^\circ)} \). Calculate \( \tan(21^\circ)\approx0.3839 \), then \( x_A\approx\frac{139}{0.3839}\approx362.07 \) feet.

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

For point B, \( \theta = 50^\circ \). So \( x_B=\frac{139}{\tan(50^\circ)} \). Calculate \( \tan(50^\circ)\approx1.1918 \), then \( x_B\approx\frac{139}{1.1918}\approx116.63 \) feet.

Step4: Find distance from A to B

Since the boat is moving towards the lighthouse, the distance from A to B is \( x_A - x_B \). So \( 362.07 - 116.63 = 245.44 \approx 245 \) feet.

Answer:

245