Chapter 4.3, Problem 68E

a.

To determine

Label the known quantities of the triangle and use a variable to represent the height of the tower.

Answer to Problem 68E

  

Explanation of Solution

Given information:

A six-foot person walks from the base of a broadcasting tower directly toward the tip of the shadow cast by the tower. When the person is  $\text{132}$ feet from the tower and $3$  feet from the tip of the shadow, the person’s shadow starts to appear beyond the tower’s shadow.

Draw a right triangle that gives a visual representation of the problem. Label the known quantities of the triangle and use a variable to represent the height of the tower.

Calculation:

The graphical represntation of the situation where $6ft$ tall man walking from base of a tower towards the tio of tower’s shadow with his shdow beginning to appear beyond the tower’s shadow when he is just $3ft$ from tip of the shadow of tower is as follows. The height of the tower is $h$ .

  

Hence, the triangle is shown above.

b.

To determine

Write an equation involving the unknown quantity.

Answer to Problem 68E

  $\boxed{h=135\times \tan \theta }$

Explanation of Solution

Given information:

A six-foot person walks from the base of a broadcasting tower directly toward the tip of the shadow cast by the tower. When the person is  $\text{132}$ feet from the tower and $3$  feet from the tip of the shadow, the person’s shadow starts to appear beyond the tower’s shadow.

Use a trigonometric function to write an equation involving the unknown quantity.

Calculation:

Assume $\angle ACB=\theta$ , then the equation involving trigonometric function is

  $\tan \theta =\frac{AB}{BC}$

  $\tan \theta =\frac{h}{135}$

  $h=135\times \tan \theta$

Hence, the equation is

  $\boxed{h=135\times \tan \theta }$

c.

To determine

What is the height of the tower?

Answer to Problem 68E

  $\boxed{h=270ft}$

Explanation of Solution

Given information:

A six-foot person walks from the base of a broadcasting tower directly toward the tip of the shadow cast by the tower. When the person is  $\text{132}$ feet from the tower and $3$  feet from the tip of the shadow, the person’s shadow starts to appear beyond the tower’s shadow.

What is the height of the tower?

Calculation:

In triangle $DEC$

  $\tan \theta =\frac{DE}{EC}$

  $\tan \theta =\frac{6}{3}$

In triangle $ABC$ , as $\theta$ is same as that in triangle $DEC$ ,

  $\tan \theta =\frac{AB}{BC}$

  $\frac{6}{3}=\frac{h}{135}$

  $h=\frac{6\times 135}{3}$

  $h=2\times 135$

  $h=271ft$

Hence, the height of the tower is

  $\boxed{h=270ft}$