Chapter 4.1, Problem 41PPS

a.

To determine

Write a quadratic function for a given situation.

Answer to Problem 41PPS

  $-x^2+6x+475$

Explanation of Solution

Given information:

Given: A babysitting club is hired for $50$ different families. They were interested to increase their current rate i.e., $\$9.50perhour$ . After observing the families, the club got to know that on an increase of $\$0.50perhour$ the number of families will decrease by $2$ .

Calculation:

Here, we will make two equations one for the fees and other for the number of families.

  $\begin{array}{l}9.5+0.5x\mathrm{...........}\left(1\right)\\ 50-2x\mathrm{...............}\left(2\right)\\ Now,multiplythefirsttermofeachequation:\\ \left(9.5\times 50\right)=475\\ Now,mutiplythefirsttermof\left(1\right)andthelasttermof\left(2\right):\\ \left(9.5\times \left(-2x\right)\right)=-19x\\ Similarly,mutiplythelasttermof\left(1\right)andthefirsttermof\left(2\right):\\ \left(0.5x\times 50\right)=25x\\ Multiplythelastterms:\\ \left(0.5x\times \left(-2x\right)\right)=-x^2\\ Now,theequationwillbe:\\ 475-19x+25x-x^2\\ -x^2+6x+475\end{array}$

Hence, the equation is $-x^2+6x+475$ .

b.

To determine

Find the domain and range of the function.

Answer to Problem 41PPS

  $0to25and0to484$

Explanation of Solution

Given information:

Given: A babysitting club is hired for $50$ different families. They were interested to increase their current rate i.e., $\$9.50perhour$ . After observing the families, the club got to know that on an increase of $\$0.50perhour$ the number of families will decrease by $2$ .

Calculation:

Here, to find the domain and range of the function we will consider the following graph:

  

Now, from the above graph we can observe that $domain>25$ and range will be also be positive as the domain includes only non-negative number of participants.

Hence, the domain and range will be $0to25and0to484$ .

c.

To determine

Find the hourly rate that will maximize the club’s income.

Answer to Problem 41PPS

  $\$3perhour$

Explanation of Solution

Given information:

Given: A babysitting club is hired for $50$ different families. They were interested to increase their current rate i.e., $\$9.50perhour$ . After observing the families, the club got to know that on an increase of $\$0.50perhour$ the number of families will decrease by $2$ .

Calculation:

Here, we have a quadratic function:

  $f\left(x\right)=-x^2+6x+475$

Now, we can see that this function is in the parabolic form:

  $\begin{array}{l}x=-\frac{b}{2a}\\ x=-\frac{6}{2\left(-1\right)}\\ x=\frac{6}{2}\\ x=3\end{array}$

Thus, $x=3$ will maximize the function.

Hence, $\$3perhour$ will maximizes the quadratic function and it is reasonable.

d.

To determine

Find the maximum income the club can make.

Answer to Problem 41PPS

  $\$484$

Explanation of Solution

Given information:

Given: A babysitting club is hired for $50$ different families. They were interested to increase their current rate i.e., $\$9.50perhour$ . After observing the families, the club got to know that on an increase of $\$0.50perhour$ the number of families will decrease by $2$ .

Calculation:

Here, we have a quadratic function:

  $f\left(x\right)=-x^2+6x+475$

Now, we can see that this function is in the parabolic form:

  $\begin{array}{l}x=-\frac{b}{2a}\\ x=-\frac{6}{2\left(-1\right)}\\ x=\frac{6}{2}\\ x=3\\ Puttingx=3in:\\ f\left(3\right)=-{\left(3\right)}^2+6\left(3\right)+475\\ =484\end{array}$

Hence, the maximum expected income is $\$484$ .