Chapter 4.1, Problem 46STP

To determine

Find the number of cookie in each package to maximize the profit.

Answer to Problem 46STP

  $3and9$

Explanation of Solution

Given information:

The Cookie Factory sells chocolate chip and peanut butter cookies in combination packages that contain between six and twelve cookies. At least three of each type of cookie should be in each package.

  $\begin{array}{lll}Cookie\hfill & Chocolatechip\hfill & Peanutbutter\hfill \\ Cost\hfill & \$0.19\hfill & \$0.13\hfill \\ \mathrm{Price}\hfill & \$0.44\hfill & \$0.39\hfill \end{array}$

Calculation:

Here, we will suppose that:

The chocolate chip cookie $=x\ge 3$

The peanut butter cookie $=y\ge 3$

Total cookies in the package:

  $x+y=\mathrm{12........}\left(1\right)$

Now, the profit will be:

  $\begin{array}{l}\mathrm{Profit}=\left(0.44-0.19\right)x+\left(0.39-0.13\right)y\\ \mathrm{Profit}=\left(0.25\right)x+\left(0.26\right)y\\ From\left(1\right):\\ y=\left(12-x\right)\\ \mathrm{Profit}=\left(0.25\right)x+\left(0.26\right)\left(12-x\right)\\ \mathrm{Profit}=3.12-\left(0.01\right)x\\ \frac{dP}{dx}=\left(-0.01\right)\end{array}$

Now, we can see that the profit is decreasing so we will check the profit at end points:

  $\begin{array}{l}{x}_{\min }=y_{\min }=3\\ x+y=12\\ \left(x,y\right)=\left(3,9\right)and\left(9,3\right)\\ P\left(3,9\right)=9.09\\ P\left(9,3\right)=3.03\\ Maximumprofit=\left(x,y\right)=\left(3,9\right)\\ \therefore Chocolatechipcookie=3\\ Peanutbuttercookie=9\end{array}$

Hence, the number of chocolate chip and peanut butter cookie is $3and9$ .