a boat is heading towards a lighthouse, whose beacon - light is 113 feet above the water. the boats crew…

Explanation:

Step1: Use the tangent function

We know that in a right - triangle (formed by the lighthouse height, the horizontal distance from the boat to the lighthouse, and the line of sight from the boat to the beacon), (\tan\theta=\frac{\text{opposite}}{\text{adjacent}}). Here, (\theta = 15^{\circ}), the opposite side (y = 113) feet (height of the lighthouse above water), and the adjacent side (x) is the horizontal distance we want to find. So, (\tan(15^{\circ})=\frac{113}{x}).

Step2: Solve for (x)

We know that (\tan(15^{\circ})=\tan(45^{\circ}- 30^{\circ})=\frac{\tan45^{\circ}-\tan30^{\circ}}{1 + \tan45^{\circ}\tan30^{\circ}}=\frac{1-\frac{\sqrt{3}}{3}}{1 + 1\times\frac{\sqrt{3}}{3}}=2-\sqrt{3}\approx0.2679). From (\tan(15^{\circ})=\frac{113}{x}), we can solve for (x) as (x=\frac{113}{\tan(15^{\circ})}). Substitute (\tan(15^{\circ})\approx0.2679) into the formula: (x=\frac{113}{0.2679}\approx421.8)

Answer:

(421.8) feet