Chapter 12.1, Problem 46PPS

a.

To determine

Draw and label a right triangle to represent the situation.

Explanation of Solution

Given:

It is given in the question that a monument casts a shadow $24$ feet long. The angle of elevation from the end of the shadow to the top of the monument is ${50}^{\circ }$ .

Graph:

A

B $24$ feet CInterpretation:

Here, draw the right triangle with AB as a monument and BC is the shadow of the monument and the angle BCD = ${50}^{\circ }$ .

b.

To determine

Find the height of the monument by using the trigonometric function.

Answer to Problem 46PPS

The height of the monument is $28.6$ feet.

Explanation of Solution

Given:

It is given in the question that a monument casts a shadow $24$ feet long. The angle of elevation from the end of the shadow to the top of the monument is ${50}^{\circ }$ .

Concept Used:

In this, use the concept of trigonometric ratios and the formula will be, $\sin \theta =\frac{p}{h},\cos \theta =\frac{b}{h},\tan \theta =\frac{p}{b},\csc \theta =\frac{h}{p},\sec \theta =\frac{h}{b},\cot \theta =\frac{b}{p}$

Calculation:

Here, use the trigonometric ratios,

  $\begin{array}{l}\tan \theta =\frac{p}{b}\\ \tan {50}^{\circ }=\frac{x}{24}\\ 1.191=\frac{x}{24}\\ x=24\cdot 1.191\\ x\approx 28.6\end{array}$

Conclusion:

The height is $28.6$ feet.

c.

To determine

Write the function to find the height of the monument.

Answer to Problem 46PPS

The function will be $\tan {50}^{\circ }=\frac{x}{24}$ .

Explanation of Solution

Given: It is given in the question that a monument casts a shadow $24$ feet long. The angle of elevation from the end of the shadow to the top of the monument is ${50}^{\circ }$ .

Concept Used:

In this, use the concept of trigonometric ratios and the formula will be, $\sin \theta =\frac{p}{h},\cos \theta =\frac{b}{h},\tan \theta =\frac{p}{b},\csc \theta =\frac{h}{p},\sec \theta =\frac{h}{b},\cot \theta =\frac{b}{p}$.

Calculation:

Here, use the trigonometric ratios,

  $\begin{array}{l}\tan \theta =\frac{p}{b}\\ \tan {50}^{\circ }=\frac{x}{24}\end{array}$

Conclusion:

The function will be $\tan {50}^{\circ }=\frac{x}{24}$ .