Question

problem #1:
the angle of elevation from a boat to the top of a lighthouse is 25°.
the lighthouse is 150 feet tall. what is the distance from the boat
to the lighthouse?
(there is an image of a lighthouse, a boat, and a right triangle with 150 ft labeled and 25° labeled, and x representing the distance from the boat to the lighthouse)

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the height of the lighthouse (opposite side to the angle of elevation) is 150 feet, the distance from the boat to the lighthouse is the adjacent side (let's call it \( x \)), and the angle of elevation is \( 25^\circ \). We use the tangent function, which is defined as \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). So, \( \tan(25^\circ)=\frac{150}{x} \).

Step2: Solve for \( x \)

Rearrange the formula to solve for \( x \): \( x = \frac{150}{\tan(25^\circ)} \). We know that \( \tan(25^\circ)\approx0.4663 \). Substitute this value into the formula: \( x=\frac{150}{0.4663}\approx321.7 \) (rounded to one decimal place).

Answer:

The distance from the boat to the lighthouse is approximately 321.7 feet.