Chapter 4.1, Problem 122E

a.

To determine

To write: the area and arc length as function of r and find the domain of each function and use the graphing utility to graph the function . Use the graph to find which changes more rapidly as r increases.

Answer to Problem 122E

  $\begin{array}{l}A=0.4r^2.\\ s=0.8r.\end{array}$

The domain of each function could be all positive real numbers.

As r increases, the area will grow more rapidly because the area formula is quadratic in r , while the arc length is linear.

Explanation of Solution

Given information: The formula for the area of a circular sector and arc length are

  $A=\frac{1}{2}{r}^2\theta \text{and}s=r\theta$ , respectively. ( r is the radius and $\theta$ is the angle measured in radians).

Let $\theta =0.8$

Calculation:

  $\begin{array}{c}For\theta =0.8.\\ A=\frac{1}{2}{r}^2(0.8)=0.4r^2.\\ s=0.8r.\end{array}$

The graph of each function is given below.

  

So, the domain of each function could be all positive real numbers.

As r increases, the area will grow more rapidly because the area formula is quadratic in r , while the arc length is linear.

b.

To determine

To write: the area and arc length as function of $\theta$and find the domain of each function and use the graphing utility to graph and identify the function.

Answer to Problem 122E

  $\begin{array}{l}A=50\theta .\\ s=10\theta .\end{array}$

So, the domain of each function could be (0, $2\pi$ ).

In this case both are linear, but the area increases at a higher rate because the slope, 50 is greater.

Explanation of Solution

Given information: The formula for the area of a circular sector and arc length are

  $A=\frac{1}{2}{r}^2\theta \text{and}s=r\theta$ , respectively. ( r is the radius and $\theta$ is the angle measured in radians).

Let $r=10\text{centimeter}$ .

Calculation:

  $\begin{array}{c}Forr=10.\\ $ A=\frac{1}{2}{(10)}^2\theta =50\theta .$s=10\theta .\end{array}$

The graph of each function is given below.

  

So, the domain of each function could be (0, $2\pi$ ).

In this case both are linear, but the area increases at a higher rate because the slope , 50 is greater.