Chapter 4.1, Problem 59PPS

a.

To determine

Write and graph the quadratic equation for price increase.

Answer to Problem 59PPS

  $P\left(x\right)=-5x^2-35x+390$

  

Explanation of Solution

Given information:

Omar sells $600$ cans of soda per week at $\$0.65$ per can. He estimated that he will lose and gain $100$ customers for every increase and decrease $\$0.05$ in price.

Calculation:

Assume $x$ be the price that maximizes the income, $P\left(x\right)$ s the function of $x$ .

  $\begin{array}{l}Amountis\$\left(0.65+0.05x\right)\\ Numberofpeopleis\left(600-100x\right)\\ P\left(x\right)=\left(0.65+0.05x\right)\left(600-100x\right)\\ =390-65x+30x-5x^2\\ =-5x^2-35x+390\end{array}$

Hence, quadratic equation is $P\left(x\right)=-5x^2-35x+390$

Now draw graph,

  

Hence, the graph plotted.

b.

To determine

Find the change in price to increase the income.

Answer to Problem 59PPS

Omar should have $3$ price decrease and charge $\$0.5$ or, $4$ price decrease and charge $\$0.45$ .

Explanation of Solution

Given information:

Omar sells $600$ cans of soda per week at $\$0.65$ per can. He estimated that he will lose and gain $100$ customers for every increase and decrease $\$0.05$ in price.

Calculation:

Assume $x$ be the price that maximizes the income, $P\left(x\right)$ s the function of $x$ .

  $\begin{array}{l}Amountis\$\left(0.65+0.05x\right)\\ Numberofpeopleis\left(600-100x\right)\\ P\left(x\right)=\left(0.65+0.05x\right)\left(600-100x\right)\\ =390-65x+30x-5x^2\\ =-5x^2-35x+390\end{array}$

Hence, quadratic equation is $P\left(x\right)=-5x^2-35x+390$

  $P\left(x\right)$ is a in parabolic form. And coefficient of $x^2$ is negative then equation of parabola must have maximum value,

Find $x$ -coordinates of vertex of parabola,

  $\begin{array}{l}x=-\frac{b}{2a}\\ =-\frac{35}{2\left(-5\right)}\\ =\frac{35}{10}\\ =3.5\end{array}$

The maximum value of function $P\left(x\right)$ attains at $x=3.5$

The required change in price that maximize the income is $\$3.5$

  $\begin{array}{l}loweramountis\$\left(0.65-0.05x\right)\\ \$\left(0.65-0.05\times 3\right)=\$0.5\\ and\\ \$\left(0.65-0.05\times 4\right)=\$0.45\end{array}$

Hence, Omar should have $3$ price decrease and charge $\$0.5$ or, $4$ price decrease and charge $\$0.45$ .

c.

To determine

Find the per week income.

Answer to Problem 59PPS

  $\$450$

Explanation of Solution

Given information:

Omar sells $600$ cans of soda per week at $\$0.65$ per can. He estimated that he will lose and gain $100$ customers for every increase and decrease $\$0.05$ in price.

Calculation:

Assume $x$ be the price that maximizes the income, $P\left(x\right)$ s the function of $x$ .

  $P\left(x\right)=-5x^2-35x+390$

Omar have $3$ price decrease and charge $\$0.5$

  $\begin{array}{l}putx=3\\ P\left(3\right)=-5\times 3^2-35\times 3+390\\ =450\end{array}$

Hence, income per week is, $\$450$ .