Question

A boat is heading towards a lighthouse, where Deion is watching from a vertical distance of 140 feet above the water. Deion measures an angle of depression to the boat at point A to be 15 degrees2

At some later time, Deion takes another measurement and finds the angle of depression to the boat (now at point B) to be 51∘
Find the distance from point A to point B. Round your answer to the nearest foot if necessary.

Answer 1

Check the picture below.

`\tan(51^o)=\cfrac{140}{x}\implies x=\cfrac{140}{\tan(51^o)} \\\\[-0.35em] ~\dotfill\\\\ \tan(15^o)=\cfrac{140}{y+x}\implies y\tan(15^o)+x\tan(15^o)=140 \\\\\\ y\tan(15^o)=140-x\tan(15^o)\implies \stackrel{\textit{substituting from above}}{y\tan(15^o)=140-\left( \cfrac{140}{\tan(51^o)} \right)\tan(15^o)} \\\\\\ y\tan(15^o)=140-\cfrac{140\tan(15^o)}{\tan(51^o)}`

`y\tan(15^o)=\cfrac{140\tan(51^o)-140\tan(15^o)}{\tan(51^o)} \\\\\\ y=\cfrac{140[\tan(51^o)-\tan(15^o)]}{\tan(15^o)\cdot \tan(51^o)}\implies \boxed{y\approx 409~ft}`

Make sure your calculator is in Degree mode.