Chapter 3, Problem 13STP

To determine

Of the given situation, we have to frame a system of inequalities and also graph it showing feasible region and coordinates. Also, we have to write a profit function for selling the number cookies and muffins and finally, calculate the maximum profit from it.

Answer to Problem 13STP

(a) The inequalities are:

  $x\ge 0$

  $y\ge 0$

  $5x+5y\le 40$

  $2x+y\le 15$

(c) Profit function = $12 x + $8 y

(d) maximumprofit = $ 92

Explanation of Solution

Given Information

ingredients for each tray of cookies = 5 cups of flour + 2 cups of sugar

ingredients for each tray of muffins = 5 cups of flour + 1 cups of sugar

Total number of cups of flour available = 40

Total number of cups of sugar available = 15

Profit per tray of cookies = $ 12

Profit per tray of muffins = $ 8

Calculation (a)

Let,

x = number of trays of cookies baked

y = number of trays of muffins baked

So, according to the data’s given,

  $x\ge 0$  ......(i)

  $y\ge 0$  ......(ii)

  $5x+5y\le 40$  ......(iii)

  $2x+y\le 15$  ......(iv)

(b) Solving equation (iii) and (iv) to intercept form, that is,

  $\frac{x}{a}+\frac{y}{b}=1$

where ‘ a’ and ‘ b’ will be ‘ x’ and ‘ y’ intercept respectively.

from (iii), we have

  ⇒ $\frac{5x}{40}+\frac{5y}{40}=1$

  ⇒ $\frac{x}{8}+\frac{y}{8}=1$

  $[a_1=8,b_1=8]$

from (iv), we have

  ⇒ $\begin{array}{l}2x+y=15\\ \frac{x}{\left(\frac{15}{2}\right)}+\frac{y}{15}=1\end{array}$

  $[a_2=\frac{15}{2},b_2=15]$

Graph

Plotting these on the graph, we get,

  

co-ordinates of vertices in feasible region,

A = (0, 8)

B = $\left(\frac{15}{2},0\right)$

P = (7, 1)

(c) Profit function = $ 12x + $ 8y

(d) Maximum profit

For maximum profit, Tonya should sell, 7 trays of cookies and 1 tray of muffins

Therefore, maximum profit = ($ 12 × 7) + ($ 8 × 1)

            = $ 84 + $ 8

            = $ 92