Chapter 8.5, Problem 14PPS

To determine

To calculate:

The name of light which will reach to boat positioned directly between the two light houses

Answer to Problem 14PPS

Little Gull Island lighthouse will reach the boat positioned directly between the two light houses.

Explanation of Solution

Given information:

Height of Little Gull Island Lighthouse: $91ft$

Angle of depression for Little Gull Island Lighthouse: $2°,6°$

Calculation:

The relationship between the angle of elevation and the angle of depression is given by alternate interior angles theorem. As per the theorem both are congruent in nature.

Consider,

The horizontal range of the light from Little Gull Island Lighthouse: $x$

The horizontal range of the light from Plum Island Light house: $y$

Let’s draw a figure to represent the given information:

  

If is a $△ABC$ right triangle with acute $\angle A$ , then the tangent of $\angle A$ is the ratio of the length of

the leg opposite $\angle A$ to the length of the leg adjacent $\angle A$ .

As per the definition of tangent:

  $\begin{array}{l}\tan 6°=\frac{91}{x}\\ x=\frac{91}{\tan 6°}\\ x\approx 865.8\end{array}$

But here the boat is $\frac{1}{2}\left(1800\right)=900ft$ away from Little Gull Lighthouse, the light from Little Gull Island will not reach the boat.

Let’s check for Plum Island Lighthouse:

  $\begin{array}{l}\tan 2°=\frac{34}{x}\\ x=\frac{34}{\tan 2°}\\ x\approx 973.6\end{array}$

Here the boat is $\frac{1}{2}\left(1800\right)=900ft$ away from Little Gull Lighthouse, the light from Little Gull Island will reach the boat.

Therefore, Little Gull Island lighthouse will reach the boat