Question

1. a cliff is 85 meters above the sea. from the top of the cliff, the angle of depression of a boat measures 43°. how far is the boat from the base of the cliff?

Explanation:

Step1: Understand the problem

We have a right triangle where the height of the cliff (opposite side to the angle of depression) is 85 meters, and we need to find the adjacent side (distance from the boat to the base of the cliff) using the tangent function since \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). The angle of depression is equal to the angle of elevation from the boat to the top of the cliff, so \(\theta = 43^\circ\).

Step2: Set up the tangent formula

Let \(x\) be the distance from the boat to the base of the cliff. Then \(\tan(43^\circ)=\frac{85}{x}\).

Step3: Solve for \(x\)

We can rearrange the formula to solve for \(x\): \(x=\frac{85}{\tan(43^\circ)}\). We know that \(\tan(43^\circ)\approx0.9325\).

Step4: Calculate the value of \(x\)

Substitute the value of \(\tan(43^\circ)\) into the formula: \(x = \frac{85}{0.9325}\approx91.15\) (rounded to two decimal places).

Answer:

The boat is approximately \(\boldsymbol{91.15}\) meters from the base of the cliff.