Question

an 80 - foot rope is attached to a boat 7 feet above the water level as shown and a boy jumps over the wake while wakeboarding. what is the distance between his hands and the water level? round to the nearest hundredth.

Explanation:

Step1: Find the angle \(\theta\)

Since \(\theta + 87^{\circ}=180^{\circ}\) (supplementary angles), then \(\theta = 180^{\circ}- 87^{\circ}=93^{\circ}\).

Step2: Use the sine function

Let \(h\) be the vertical distance from the boy's hands to the water. We know that \(\sin\theta=\frac{h}{80}\) (where the rope length is the hypotenuse of a right - triangle - like situation). Substituting \(\theta = 93^{\circ}\), we get \(h = 80\times\sin(93^{\circ})\).
We know that \(\sin(93^{\circ})\approx0.9986\). Then \(h = 80\times0.9986=79.89\) (rounded to the nearest hundredth). But we need to subtract the height of the boat above water. Let \(d\) be the distance between his hands and the water level. \(d=79.89 - 7\).

Step3: Calculate the final distance

\(d=79.89-7 = 72.89\)

Answer:

\(72.89\)