Chapter 9.1, Problem 10GP

Solution

To estimate: the height of the parasailer above the boat if the angle of elevation is $38°$ .

Nearly 185 ft

Given information:

A parasailer is attached to a boat with a rope 300 feet long. The angle of elevation from the boat to the parasailer is $38°$ .

Definition Used:

Let $\theta$ be an acute angle of a right triangle. The six trigonometric functions of $\theta$ are defined as follows:

  $\sin \theta =\frac{opposite}{hypotenuse}$ , $\cos \theta =\frac{adjacent}{hypotenuse}$ , $\tan \theta =\frac{opposite}{adjacent}$

  $\cot \theta =\frac{adjacent}{opposite}$ , $\csc \theta =\frac{hypotenuse}{opposite}$ , $\sec \theta =\frac{hypotenuse}{adjacent}$

Theorem Used:

Pythagorean Theorem:

In a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse.

  $Hypotenus{e}^2=opposit{e}^2+adjacen{t}^2$

For the given triangle, this is:

  

  $a^2+b^2=c^2$

Explanation:

The given situation can be represented by the triangle,

  

Let the height of the parasailer above the boat be h feet. Then,

  $\begin{array}{c}\sin 38=\frac{opposite}{hypotenuse}\\ \sin 38°=\frac{h}{300}\\ h=300\times \sin 38°\\ h\approx 185\text{ feet}\end{array}$

The height of the parasailer above the boat is approximately 185 feet.