Chapter 4.8, Problem 24E

a.

To determine

Label the known and unknown quantities.

Answer to Problem 24E

  $\boxed{AB=\text{12}\frac{1}{2}ft}$ , $\boxed{BC=17\frac{1}{3}ft}$

Explanation of Solution

Given information:

The height of an outdoor basketball backboard is $\text{12}\frac{1}{2}$ feet, and the backboard casts a shadow $\text{17 feet}$ long.

Draw a right triangle that gives a visual representation of the problem. Label the known and unknown quantities.

Calculation:

Let us consider the height of the basketball backboard as $AB$ and its shadow as $BC$ .

Therefore the given problem can be represented as the right angled triangle.

The figure showing the basketball backboard and its shadow are shown below. The height of the backboard will be taken as $AB=\text{12}\frac{1}{2}ft$ and its shadow will be represented by $BC=17\frac{1}{3}ft$ .

We will assume the base angle of the triangle as $\theta$ .

Hence , the known and unknown quantities are $AB=\text{12}\frac{1}{2}ft$ and $BC=17\frac{1}{3}ft$ .

b.

To determine

Write an equation involving the unknown angle of elevation.

Answer to Problem 24E

  $\boxed{\theta =Arc\tan \left(0.72254\right)}$

Explanation of Solution

Given information:

The height of an outdoor basketball backboard is $\text{12}\frac{1}{2}$ feet, and the backboard casts a shadow $\text{17 feet}$ long.

Use a trigonometric function to write an equation involving the unknown angle of elevation.

Calculation:

Let us consider the height of the basketball backboard as $AB$ and its shadow as $BC$ .

Therefore the given problem can be represented as the right angled triangle.

We will use the fact that $AB=\text{12}\frac{1}{2}ft$ and $BC=17\frac{1}{3}ft$ .

Now to find the angle of elevation $\theta$ we will apply the trigonometric identity

  $\tan \theta =\frac{\text{Opposite}}{\text{Hypotenuse}}$

  $=\frac{AB}{BC}$

  $=\frac{12\frac{1}{2}}{17\frac{1}{3}}$

  $=0.72254$

  $\theta =Arc\tan \left(0.72254\right)$

Hence, the unknown angle of elevation is $\theta =Arc\tan \left(0.72254\right)$

c.

To determine

Find the angle of elevation.

Answer to Problem 24E

  $\boxed{35.85"}$

Explanation of Solution

Given information:

The height of an outdoor basketball backboard is $\text{12}\frac{1}{2}$ feet, and the backboard casts a shadow $\text{17 feet}$ long.

Find the angle of elevation.

Calculation:

Let us consider the height of the basketball backboard as $AB$ and its shadow as $BC$ .

Therefore the given problem can be represented as the right angled triangle

As calculated in part (b), we can say that the angle of elevation to the sun is given by

Hence, $Arc\tan \left(0.72254\right)=\boxed{35.85"}$ .