Chapter 6.1, Problem 62E

(a)

To determine

To Find: The area of the shaded region in the given figure 1

Answer to Problem 62E

The area of the shaded region is $300\sin \frac{3\theta }{2}-120\sin \theta -80\sin \frac{\theta }{2}$ .

Explanation of Solution

Given:

The given figure is shown in Figure 1

  

Figure 1

Calculation:

Consider the area of the shaded region from the figure 1 is calculated as,

  $\begin{array}{c}{A}_{ABC}=\frac{bc\sin A}{2}\\ =\frac{AB\cdot AC\cdot \sin \angle BAC}{2}\\ =\frac{30\cdot 20\cdot \sin \left(\theta +\frac{\theta }{2}\right)}{2}\\ =300\sin \frac{3\theta }{2}\end{array}$

Consider the area of the triangle $A_{ABD}$ is obtained as,

  $\begin{array}{c}{A}_{ABD}=\frac{AC\cdot AD\cdot \sin \angle BAD}{2}\\ =\frac{30\cdot 8\sin \theta }{2}\\ =120\sin \theta \end{array}$

Consider the area of the triangle $A_{ACD}$ is obtained as,

  $\begin{array}{c}{A}_{ACD}=\frac{AC\cdot AD\cdot \sin \angle DAC}{2}\\ =\frac{20\cdot 8\sin \frac{\theta }{2}}{2}\\ =80\sin \frac{\theta }{2}\end{array}$

Then, the triangle $A_{CBD}$ is obtained as,

  $\begin{array}{c}{A}_{CBD}=A_{ABC}-A_{ABD}-A_{ACD}\\ =300\sin \frac{3\theta }{2}-120\sin \theta -80\sin \frac{\theta }{2}\end{array}$

(b)

To determine

To Find: The graphing utility to graph the function.

Answer to Problem 62E

The required graph is shown in Figure 2

Explanation of Solution

Consider the required function is,

  $A_{CBD}=300\sin \frac{3\theta }{2}-120\sin \theta -80\sin \frac{\theta }{2}$

The graph for the above triangle is shown in Figure 2

  

Figure 2

(c)

To determine

To Find: The domain of the function and explain how the decreasing length of the eight centimetre line segment affects the area of the region and the domain of the function.

Answer to Problem 62E

The domain of the function is in the range $0\le \theta \le 1.66$ . Also, the decreasing length of the eight centimetre line as the length of the AD increase the area of the shaded region decreases.

Explanation of Solution

Consider the graph for the function shown in Figure 3

  

Figure 3

From the above figure the domain of the function is in the range $0\le \theta \le 1.66$ .

Consider the diagram shown in Figure 4

  

Figure 4

The figure shows that as the length of the AD increase the area of the shaded region decreases. For the line segment of the length decreasing the domain increases. The negative part of the area expression decreases as it attains the maximum value.