Chapter 6.1, Problem 61E

a.

To determine

Write the area $A$ of the shaded region in the figure as a function of $\theta$ .

Answer to Problem 61E

  $=\boxed{20\left(15\sin \frac{3\theta }{2}-6\sin \theta -4\sin \frac{\theta }{2}\right)}$

Explanation of Solution

Given information:

Write the area $A$ of the shaded region in the figure as a function of $\theta$ .

  

Calculation:

Let us consider the following figure.

  

Let A be the area of the shaded region in the above figure.

If $ABC$ is a triangle with sides $a,b$ and $c$ .then the area of the triangle is

  $Area=\frac{1}{2}bc\sin A$

  $\begin{array}{l}=\frac{1}{2}a\sin B\\ =\frac{1}{2}ab\sin C\end{array}$

Hence, we get

  $A=\frac{1}{2}\left(30\right)\left(20\right)\sin \frac{3\theta }{2}-\frac{1}{2}\left(30\right)\left(8\right)\sin \theta -\frac{1}{2}\left(8\right)\left(20\right)\sin \frac{\theta }{2}$

  $=\boxed{20\left(15\sin \frac{3\theta }{2}-6\sin \theta -4\sin \frac{\theta }{2}\right)}$

b.

To determine

Plot the graph.

Answer to Problem 61E

  

Explanation of Solution

Given information:

Use a graphing utility to graph the function.

Calculation:

From the part (a) we have

  $A=120\sin \theta$

Now by uing graphing utility we get the graph of the above equation.

  $>smartplot[x,y](y=120\sin (x))$

  

c.

To determine

Determine the domain of the function.

Answer to Problem 61E

  $\boxed{0\le \theta \le 1.66}$

Explanation of Solution

Given information:

Determine the domain of the function. Explain how decreasing the length of the eight-centimetre line segment would affect the area of the region and the domain of the function.

Calculation:

From the part (a) we have

  $A=20\left(15\sin \frac{3\theta }{2}-6\sin \theta -4\sin \frac{\theta }{2}\right)$

Now from the graph we can see that the domain of the function is $\boxed{0\le \theta \le 1.66}$ . When the line seg,ent of length eight-centimeter were decreasing than the domain that means value of $\theta$ increases. Also the neagtive part of area expression decreases that means area attains its maximum value.

Hence,the domain of the function is $\boxed{0\le \theta \le 1.66}$ .