Chapter MPS, Problem 5.1MPS

(a)

To determine

To find: The charge for each mug to maximize the monthly income.

Answer to Problem 5.1MPS

  $\$5$

Explanation of Solution

Given:

  $\begin{array}{l}\text{Coffee mugs sold per month = 300}\\ \text{price of coffee mugs initially = \$7}\text{.50}\\ \text{Price incresed per coffee mug = \$0}\text{.75}\end{array}$

Formula Used:

  $\begin{array}{l}y\left(x\right)=a{x}^2+bc+c\\ \text{and for the parabola vertex is at }x=\frac{-b}{2a}\end{array}$

Calculation:

  $\begin{array}{l}\text{if the price is increased by }\$0.75x\\ \text{then}\\ \text{no}\text{. of coffee mugs sold are }300-15x\\ \text{so the income is }\\ I\left(x\right)=\left(7.50+.75x\right)\left(300-15x\right)\\ \text{⇒}-11.25x^2+112.5x+2250\\ \text{since the equation is of a parabola}\\ \text{the vertex is }\\ \text{⇒}x=\frac{-b}{2a}\\ \text{⇒}x=\frac{-112.5}{2\left(-11.25\right)}=\frac{112.5}{\begin{array}{l}22.5\end{array}}\\ \text{⇒}x=\$5\end{array}$

Conclusion:

Thus, the mug cost is to be $\$5$ .

(b)

To determine

To Find: The maximum monthly income of the owner from the mugs.

Answer to Problem 5.1MPS

  $\$2531.25$

Explanation of Solution

Given:

  $\begin{array}{l}\text{Coffee mugs sold per month = 300}\\ \text{price of coffee mugs initially = \$7}\text{.50}\\ \text{Price incresed per coffee mug = \$0}\text{.75}\end{array}$

Formula Used:

  $\begin{array}{l}y\left(x\right)=a{x}^2+bc+c\\ \text{and for the parabola vertex is at }x=\frac{-b}{2a}\end{array}$

Calculation:

  $\begin{array}{l}\text{if the price is increased by }\$0.75x\\ \text{then}\\ \text{no}\text{. of coffee mugs sold are }300-15x\\ \text{so the income is }\\ I\left(x\right)=\left(7.50+.75x\right)\left(300-15x\right)\\ \text{⇒}-11.25x^2+112.5x+2250\\ \text{since the equation is of a parabola}\\ \text{the vertex is }\\ \text{⇒}x=\frac{-b}{2a}\\ \text{⇒}x=\frac{-112.5}{2\left(-11.25\right)}=\frac{112.5}{\begin{array}{l}22.5\end{array}}\\ \text{⇒}x=\$5\\ \text{so the maximum income is at }x=5\\ I\left(5\right)=-11.25\left(5^2\right)+112.5\left(x\right)+2250\\ I\left(5\right)=\$2531.25\end{array}$

Conclusion:

Thus the maximum monthly income will be $\$2531.25$ .