Chapter 3.2, Problem 24PPS

To determine

To calculate: The combinations of batches of each could Rebecca make.

Answer to Problem 24PPS

  $(20,5)$ (20 batches of the cookies,5 batches of cupcakes)

  $(19,6)$ (19 batches of the cookies,6 batches of cupcakes) or,

  $(23,1)$ (23 batches of the cookies,1 batches of cupcakes)

Explanation of Solution

Given information:

Rebecca can cook 15 cookies and 12 cupcakes at a time.

Minimum goods she had to cook = 120

Maximum goods she can cook = 360

Calculation:

Let,

X=number of cookie batches (1 batch=15cookies)

Y=number of cupcake batches (1 batch=12 cupcakes)

At least 120 but no more than 360 total cookies and cupcakes

  $120\le 15x+12y\le 360$

At least 3 times as many cookies as cupcakes

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

Isolate the y in the middle

  $\begin{array}{l}120\le 15x+12y\le 360\\ 120-15x\le 12y\le 360-15y\end{array}$

  $10-\frac{5}{4}x\le y\le 30-\frac{5}{4}x$ (red in graph)

The solution region to the inequality is the area between the boundary lines

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

Draw as solid lines since the inequality is “or equal to”

For $y\le \frac{5}{12}$ (blue in graph)

Draw the boundary line $y=\frac{5}{12}$ x as solid line. For an inequality “ $y\le \mathrm{...}$ ”

The solution region is the area below and including the boundary line.

The passible combinations of batches are the whole combinations of (x, Y) in the region where both inequalities overlap. Such as-

  $(20,5)$ (20 batches of the cookies,5 batches of cupcakes)

  $(19,6)$ (19 batches of the cookies,6 batches of cupcakes)

  $(23,1)$ (23 batches of the cookies,1 batches of cupcakes)