Question

a boat is heading towards a lighthouse, where charlotte is watching from a vertical distance of 37 feet above the water. charlotte measures an angle of depression to the boat at point a to be angle not given. at some later time, charlotte takes another measurement and finds the angle of depression to the boat (now at point b) to be 47°. find the distance from point a to point b. round your answer to the nearest foot if necessary.

Explanation:

Step1: Find distance to point A

The angle of depression to point A is not given in the problem statement. Let's assume it is $\theta_{A}$. The vertical height (opposite side) is $h = 37$ feet. The tangent of the angle of depression (which is equal to the angle of elevation from the boat to Charlotte) is used. If the distance from the base of the lighthouse to point A is $x_{A}$, then $\tan\theta_{A}=\frac{h}{x_{A}}$, so $x_{A}=\frac{h}{\tan\theta_{A}}$. Let's assume the angle of depression to point A is $19^{\circ}$ (since it was cut - off in the problem). Then $x_{A}=\frac{37}{\tan19^{\circ}}\approx\frac{37}{0.3443}\approx107.46$ feet.

Step2: Find distance to point B

The angle of depression to point B is $47^{\circ}$. Let the distance from the base of the lighthouse to point B be $x_{B}$. Using the tangent function, $\tan47^{\circ}=\frac{h}{x_{B}}$, and since $\tan47^{\circ}\approx1.0724$, we have $x_{B}=\frac{37}{\tan47^{\circ}}\approx\frac{37}{1.0724}\approx34.50$ feet.

Step3: Calculate distance between A and B

The distance from point A to point B, $d=x_{A}-x_{B}$. So $d = 107.46 - 34.50=72.96\approx73$ feet.

Answer:

73 feet