Chapter 6.3, Problem 23E

a.

To determine

To find:Type of function for the number of donations collected by organization $A,B\&C$ .

Answer to Problem 23E

Number of donation collected as function by organization $A$ is exponential, organization $B$ is linear & organization $C$ is exponential.

Explanation of Solution

Given:

All three organization are collecting donations for a cause.

Concept Used:

Use patterns between consecutive data pairs to determine type of function. Linear function: - The difference of consecutive y-values is constant. Exponential function: - Consecutive y-values have a common ratio. In each case, the differences of consecutive x-values need to be constant.

Calculation:

Linear function: - The difference of consecutive y-values is constant. Exponential function: - Consecutive y-values have a common ratio. In each case, the differences of consecutive x-values need to be constant.

Table &Graph given

  

Organization $A$ number of donations quadruples each hour, organization $B$ table given &organization $C$ graph provided.

We determine function of organization $A$ as a function of hours therefore, function is exponential.

  $\begin{array}{l}f(x)=a{b}^x\\ a=1\\ b=4\\ f(x)=4^x\end{array}$

According to the graph of organization $B$ each time there is an increase of hours, number of donation growth by a constant amount, so function is linear.

  $\begin{array}{l}g(x)=mx+n\\ g(0)=0\Rightarrow m(0)+n\Rightarrow g(x)=mx\\ x=1\Rightarrow g(1)=4\Rightarrow m=4\\ g(x)=4x\end{array}$

We determine the exponential function for organization $C$ .

  $\begin{array}{l}h(x)=p{q}^x\\ h(0)=1\Rightarrow p{q}^0=1\Rightarrow p=1\\ h(1)=3\Rightarrow p{q}^1=3\Rightarrow q=3\\ h(x)=3^x\end{array}$

Hence,

Number of donation collected as function by organization $A$ is exponential, organization $B$ is linear & organization $C$ is exponential.

b.

To determine

To find:Average rate of change of each function for each $1$ hour interval from $t=0$ to $t=6$ .

Answer to Problem 23E

Average rates of change for each function calculated.

Explanation of Solution

Given:

All three organization are collecting donations for a cause.

Concept Used:

Use patterns between consecutive data pairs to determine type of function. Linear function: - The difference of consecutive y-values is constant. Exponential function: - Consecutive y-values have a common ratio. In each case, the differences of consecutive x-values need to be constant.

Calculation:

Linear function: - The difference of consecutive y-values is constant. Exponential function: - Consecutive y-values have a common ratio. In each case, the differences of consecutive x-values need to be constant.

Table & Graph given

  

Organization $A$ number of donations quadruples each hour, organization $B$ table given &organization $C$ graph provided.

We determine the average rates of change for each function: -

We determine function of organization $A$ as a function of hours therefore, function is exponential.

  $f(x)=4^x$

  $\begin{array}{l}\frac{f(1)-f(0)}{1-0}=4-1=3\\ \frac{f(2)-f(1)}{2-1}=16-4=12\\ \frac{f(3)-f(2)}{3-2}=64-16=48\\ \frac{f(4)-f(3)}{4-3}=256-64=192\\ \frac{f(5)-f(4)}{5-4}=1024-256=768\\ \frac{f(6)-f(5)}{6-5}=4096-1024=3072\end{array}$

According to the graph of organization $B$ each time there is an increase of hours, number of donation growth by a constant amount, so function is linear.

  $g(x)=4x$

  $\begin{array}{l}\frac{g(1)-g(0)}{1-0}=4-0=4\\ \frac{g(2)-g(1)}{2-1}=8-4=4\\ \frac{g(3)-g(2)}{3-2}=12-8=4\\ \frac{g(4)-g(3)}{4-3}=16-12=4\\ \frac{g(5)-g(4)}{5-4}=20-16=4\\ \frac{g(6)-g(5)}{6-5}=24-20=4\end{array}$

We determine the exponential function for organization $C$ .

  $h(x)=3^x$

  $\begin{array}{l}\frac{h(1)-h(0)}{1-0}=3-1=2\\ \frac{h(2)-h(1)}{2-1}=9-3=6\\ \frac{h(3)-h(2)}{3-2}=27-9=18\\ \frac{h(4)-h(3)}{4-3}=81-27=54\\ \frac{h(5)-h(4)}{5-4}=243-81=162\\ \frac{h(6)-h(5)}{6-5}=729-243=486\end{array}$

Hence,

Average rates of change for each function calculated.

v.

To determine

To find:Which function does average rate of change increase most quickly.

Answer to Problem 23E

The average rate of change increases more quickly for the function $f(x)=4^x$ . Organization $A$ number of donations collected more is same no. of hours.

Explanation of Solution

Given:

All three organization are collecting donations for a cause.

Concept Used:

Use patterns between consecutive data pairs to determine type of function. Linear function: - The difference of consecutive y-values is constant. Exponential function: - Consecutive y-values have a common ratio. In each case, the differences of consecutive x-values need to be constant.

Calculation:

Linear function: - The difference of consecutive y-values is constant. Exponential function: - Consecutive y-values have a common ratio. In each case, the differences of consecutive x-values need to be constant.

Table & Graph given

  

Organization $A$ number of donations quadruples each hour, organization $B$ table given &organization $C$ graph provided.

We determine the average rates of change for each function: -

We determine function of organization $A$ as a function of hours therefore, function is exponential.

  $f(x)=4^x$

  $\begin{array}{l}\frac{f(1)-f(0)}{1-0}=4-1=3\\ \frac{f(2)-f(1)}{2-1}=16-4=12\\ \frac{f(3)-f(2)}{3-2}=64-16=48\\ \frac{f(4)-f(3)}{4-3}=256-64=192\\ \frac{f(5)-f(4)}{5-4}=1024-256=768\\ \frac{f(6)-f(5)}{6-5}=4096-1024=3072\end{array}$

According to the graph of organization $B$ each time there is an increase of hours, number of donation growth by a constant amount, so function is linear.

  $g(x)=4x$

  $\begin{array}{l}\frac{g(1)-g(0)}{1-0}=4-0=4\\ \frac{g(2)-g(1)}{2-1}=8-4=4\\ \frac{g(3)-g(2)}{3-2}=12-8=4\\ \frac{g(4)-g(3)}{4-3}=16-12=4\\ \frac{g(5)-g(4)}{5-4}=20-16=4\\ \frac{g(6)-g(5)}{6-5}=24-20=4\end{array}$

We determine the exponential function for organization $C$ .

  $h(x)=3^x$

  $\begin{array}{l}\frac{h(1)-h(0)}{1-0}=3-1=2\\ \frac{h(2)-h(1)}{2-1}=9-3=6\\ \frac{h(3)-h(2)}{3-2}=27-9=18\\ \frac{h(4)-h(3)}{4-3}=81-27=54\\ \frac{h(5)-h(4)}{5-4}=243-81=162\\ \frac{h(6)-h(5)}{6-5}=729-243=486\end{array}$

Hence,

The average rate of change increases more quickly for the function $f(x)=4^x$ . Organization $A$ number of donations collected more is same no. of hours.