Chapter 4.3, Problem 83E

(a).

To determine

To draw:

A right triangle that gives a visual representation of the problem.

Explanation of Solution

Given:

A six-foot person walks from the base of a streetlight directly toward the tip of the shadow cast by the streetlight. When the person is $16$ feet from the streetlight and $5$ feet from the tip of the streetlight’s shadow, the person’s shadow starts to appear beyond the streetlight’s shadow.

Draw:

Use the given formula and draw a right triangle.

  

(b).

To determine

To find:

An equation involving the unknown quantity.

Answer to Problem 83E

The equation involving the unknown quantity is $\tan \alpha =\frac{h}{21}$ .

Explanation of Solution

Given:

A six-foot person walks from the base of a streetlight directly toward the tip of the shadow cast by the streetlight. When the person is $16$ feet from the streetlight and $5$ feet from the tip of the streetlight’s shadow, the person’s shadow starts to appear beyond the streetlight’s shadow.

Calculation:

  

Apply the formula for the tangent trigonometric ratio:

  $\tan \alpha =\frac{P}{B}$

Substitute $P=h$ and $B=21\left(16+5\right)$ in the above formula:

  $\tan \alpha =\frac{h}{21}$

Conclusion:

Hence, the required equation involving the unknown quantity is $\tan \alpha =\frac{h}{21}$ .

(c).

To determine

To find:

The height of the streetlight.

Answer to Problem 83E

The height of the streetlight is $25.2\text{ ft}$ .

Explanation of Solution

Given:

A six-foot person walks from the base of a streetlight directly toward the tip of the shadow cast by the streetlight. When the person is $16$ feet from the streetlight and $5$ feet from the tip of the streetlight’s shadow, the person’s shadow starts to appear beyond the streetlight’s shadow.

  

From the small triangle:

  $\tan \alpha =\frac{6}{5}\mathrm{........}\left(1\right)$

Also, from the second part:

  $\tan \alpha =\frac{h}{21}\mathrm{........}\left(2\right)$

Equate equation $\left(1\right)$ and $\left(2\right)$ .

  $\begin{array}{c}\frac{h}{21}=\frac{6}{5}\\ h=\frac{6}{5}\left(21\right)\\ h=25.2\text{ ft}\end{array}$

Hence, the height of the streetlight is $25.2\text{ ft}$ .