Chapter 4.8, Problem 48E

Solution

a.

The height BE .

The height BE is approximately 23.84 ft

Given:

Given, a roof has two sections, one with a 50° elevation and the other with a 20° elevation as shown in the figure below:

  

Calculation:

Consider the triangle ABE .

Using the trigonometric function definitions,

  $\begin{array}{c}\tan A=\frac{\text{opposite side}}{\text{adjacent side}}\\ \tan \left(50°\right)=\frac{BE}{AB}\\ BE=20\left(1.192\right)\\ BE\approx 23.84\end{array}$

Conclusion:

The height BE is approximately 23.84 ft

b.

The height CD .

The height CD is approximately 40.22 ft

Given:

Given, a roof has two sections, one with a 50° elevation and the other with a 20° elevation as shown in the figure below:

  

Calculation:

Consider the triangle EOD .

Using the trigonometric function definitions,

  $\begin{array}{c}\tan E=\frac{\text{opposite side}}{\text{adjacent side}}\\ \tan \left(20°\right)=\frac{OD}{EO}\\ OD=45\left(0.364\right)\\ OD\approx 16.38\end{array}$

From the figure,

  $\begin{array}{c}CD=CO+OD\\ CD=BE+OD\\ CD=23.84+16.38\\ CD=40.22\end{array}$

Conclusion:

The height CD is approximately 40.22 ft

c.

The length AE + ED , and to double it to find the length of the roofline.

The length AE + ED is approximately 79 ft, and the length of the roofline is approximately 158 ft.

Given:

Given, a roof has two sections, one with a 50° elevation and the other with a 20° elevation as shown in the figure below:

  

Calculation:

Consider the triangle ABE .

Using the trigonometric function definitions,

  $\begin{array}{c}\cos A=\frac{\text{adjacent side}}{\text{hypotenuse}}\\ \cos \left(50°\right)=\frac{AB}{AE}\\ AE=\frac{20}{0.643}\\ AE\approx 31.11\end{array}$

Consider the triangle EOD .

Using the trigonometric function definitions,

  $\begin{array}{c}\cos E=\frac{\text{adjacent side}}{\text{hypotenuse}}\\ \cos \left(20°\right)=\frac{EO}{ED}\\ ED=\frac{45}{0.939}\\ ED\approx 47.89\end{array}$

Hence, $AE+ED=31.11+47.89=79$

Therefore, the length of the roofline is $2\left(AE+ED\right)=2\left(79\right)=158$

Conclusion:

The length AE + ED is approximately 79 ft, and the length of the roofline is approximately 158 ft.