Question

a boat is tied to the dock. what is the length of the string? 24 16° x

Explanation:

Step1: Identify the trigonometric ratio

We have a right triangle where the adjacent side to the \(16^\circ\) angle is 24, and the hypotenuse (length of the string) is \(x\). The cosine function relates the adjacent side and the hypotenuse: \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\).
So, \(\cos(16^\circ)=\frac{24}{x}\).

Step2: Solve for \(x\)

Rearrange the formula to solve for \(x\): \(x = \frac{24}{\cos(16^\circ)}\).
Calculate \(\cos(16^\circ)\approx0.9613\).
Then, \(x=\frac{24}{0.9613}\approx24.97\).

Answer:

The length of the string is approximately \(25\) (or more precisely, approximately \(24.97\)).