Chapter 13.6, Problem 47PPSE

Solution

a.

The graph of the function $y=4\tan \left(\theta \right)$ and the height of one triangle when $\theta =60°$ .

The graph of the given function is drawn and the height of one triangle is 6.9 ft.

Given:

The height of the triangle is:

  $y=4\tan \left(\theta \right)$

Calculation:

The graph of the function $y=4\tan \left(\theta \right)$ on a graphing calculator is shown below:

  

As per the given problem

  $\theta =60°$

Substitute the value of $\theta$ in the given function of height $y=4\tan \left(\theta \right)$ , we get

  $\begin{array}{c}y=4\tan \left(60°\right)\\ =4\sqrt{3}\\ =4\times 1.732\\ y\simeq 6.9\end{array}$

Thus, the height of the triangle is 6.9 ft.

Conclusion:

The graph of the given function is drawn and the height of the triangle is 6.9 ft.

b.

The area of one triangle when $\theta =60°$ .

The area of one triangle is 27.6 ft².

Given:

The base of the triangle is 8 ft.

From part (a), the height of the triangle when $\theta =60°$ is 6.9 ft.

Concept used:

Area of an isosceles triangle is $A=\frac{1}{2}bh$

Where, b is the side and h is the height of the triangle.

Calculation:

Area of an isosceles triangle is $A=\frac{1}{2}bh$

On substituting the values of b and h in the formula of area,

  $\begin{array}{c}A=\frac{1}{2}\times 8\times 6.9\\ A=27.6{\text{ ft}}^{\text{2}}\end{array}$

Thus, the area of one triangle is 27.6 ft².

Conclusion:

The area of one triangle is 27.6 ft².

c.

The area of the gazebo floor in square feet.

The area of the gazebo floor is 165.6 ft².

Given:

From part (b), the area of one triangle is 27.6 ft²

Calculation:

The area of the hexagonal gazebo floor can be calculated by multiplying the area of triangle by 6 as there are 6 isosceles triangles.

Thus, Area of gazebo floor is

  $\begin{array}{c}{A}_{\text{total}}=\text{Area of triangle}\times 6\\ =27.6\times 6\\ A_{\text{total}}=165.6{\text{ ft}}^2\end{array}$

Conclusion:

The area of the gazebo floor is 165.6 ft².