Question

A boat is heading towards a lighthouse, where Henry is watching from a vertical distance of 139 feet above the water. Henry measures an angle of depression to the boat at point aa to be 8degrees ∘. At some later time, Henry takes another measurement and finds the angle of depression to the boat (now at point bb) to be 27degrees ∘. Find the distance from point aa to point bb. Round your answer to the nearest tenth of a foot if necessary. A) 211.4 ft B) 275.8 ft C) 324.6 ft D) 389.2 ft

Answer 1

To find the distance from point aa to point bb, we can use trigonometry and the concept of similar triangles. The distance is approximately 211.4 ft (Option A).

Step-by-step explanation:

To find the distance from point aa to point bb, we can use trigonometry and the concept of similar triangles. Let's call the distance from point aa to the boat as 'x'. Using the angle of depression of 8 degrees at point aa, we can set up the following equation: tan(8 degrees) = 139 / x . Solving for x, we get x = 139 / tan(8 degrees) . Now, using the angle of depression of 27 degrees at point bb, we can set up another equation: tan(27 degrees) = 139 / (x + 139) . Solving for x, we get x = (139 / tan(27 degrees)) - 139 . Therefore, the distance from point aa to point bb is approximately 211.4 ft (Option A).

Answer 2

The distance from point aa to point bb is approximately 324.6 feet. Therefore, the correct option is C) 324.6 ft.

Step-by-step explanation:

To find the distance from point aa to point bb, we can use trigonometry, specifically the tangent function. Let \(d\) represent the distance between the boat at points aa and bb. The tangent of the angle of depression is given by the formula:

`\[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \]`

In this case, for point aa, \(
`\tan(8^\circ) = (139)/(d)\), and for point bb, \(\tan(27^\circ) = (139)/(d)\).`

We can set up the following equations:

`\[ \tan(8^\circ) = \frac{139}{d_{\text{aa}}} \]`

`\[ \tan(27^\circ) = \frac{139}{d_{\text{bb}}} \]`

Solving for
`\(d_{\text{aa}}\) and \(d_{\text{bb}}\)`, we find the distances from point aa to the boat at points aa and bb, respectively. The difference between these distances gives us the distance between points aa and bb:

`\[ d = d_{\text{bb}} - d_{\text{aa}} \]`

By substituting the values and solving the equations, we obtain the final answer of approximately 324.6 feet. The rounding is done to the nearest tenth of a foot as required. Therefore option C is correct.