Chapter 3.3, Problem 24PPS

To determine

To find:The combination of cookies and cupcakesfor batches Rebecca make.

Answer to Problem 24PPS

  $\left(20,5\right)$ , where twenty batch of cookies and five batch of cupcakes.

  $\left(19,6\right)$ ,where nineteen batch of cookies and 6 batch of cupcakes.

  $\left(23,1\right)$ ,where twenty-three batch of cookies a one batch of cupcakes.

Explanation of Solution

Given:It is given that Rebecca can back 15 cookies and 12 cupcakes at a time.

Rebecca needs to make at least $120$ backed goods but no more then $360$.

But she wants to have at least three times as many cookies as cupcakes.

Let $x$ be the number of cookie batchand $y$ be no. of cupcake batch. Given in question at least $120$ but no more then $360$ total cookies and cupcakes

  $120\le 15x+12y\le 360$

Given in question at least three times as many cookies as cupcakes

  $\begin{array}{l}15x\ge 3\times 12y\\ 15x\ge 36y\\ \frac{15}{36}x\ge y\\ \frac{5}{12}x\ge y\end{array}$

Isolate the $y$ in the middle

  $\begin{array}{l}120\le 15x=12y\le 360\\ 120-15x\le y\le 30-\frac{5}{4}x\\ 10-\frac{5}{4}x\le y\le 30-\frac{5}{4}x\end{array}$

  $y=10-\frac{5}{4}x$ and $y=30-\frac{5}{4}x$

Draw as solid line.

For $y\le \frac{5}{12}x$

Draw the boundary line $y=\frac{5}{12}x$ as a solid line.

  

  $\left(20,5\right)$ , where twenty batch of cookies and five batch of cupcakes.

  $\left(19,6\right)$ ,where nineteen batch of cookies and 6 batch of cupcakes.

  $\left(23,1\right)$ ,where twenty-three batch of cookies a one batch of cupcakes.