Chapter 3.3, Problem 25PPS

a)

To determine

To form: a system of inequalities performed by Sean to paint sheds and play houses.

Answer to Problem 25PPS

(a) Inequalities are:

  $\begin{array}{l}4x+5y\le 200\\ x+y\le 45\\ x\ge 0\\ y\ge 0\end{array}$

(c) 25 sheds and 20 play houses

(d) So, maximum profit= $1250

Explanation of Solution

Given Information

Let the numbers of sheds be ‘ x’ and the number of play houses be ‘ y’ .

Time required to paint 1 shed = 2.5 days and the required to paint 1 playhouse = 2 days

But, Sean has a total of 20 days to paint

So,

  $\frac{x}{2.5}+\frac{y}{2}\le 20$

  ⇒ $10\times \frac{x}{2.5}+10\times \frac{y}{2}\le 20\times 10$

  ⇒ $4x+5y\le 200$  ......(i)

Converting it in intercept form, $\left\{\frac{x}{a}+\frac{y}{b}=1\right\}$ , where ‘a’ and ‘b’ are ‘ x’and ‘ y’ intercepts respectively

  $\frac{4x}{200}+\frac{5y}{200}=1$

  ⇒ $\frac{x}{50}+\frac{y}{40}=1$  ......(ii)

  $[a_1=50,b_1=40]$

But, it is also given that, in total there are 45 structures,

So,

  $x+y\le 45$

Converting it in intercept form,

  $\frac{x}{45}+\frac{y}{45}=1$$\left\{\frac{x}{a}+\frac{y}{b}=1\right\}$

  $[a_2=45,b_2=45]$

Hence, our final inequalities are:

  $\begin{array}{l}4x+5y\le 200\\ x+y\le 45\\ x\ge 0\\ y\ge 0\end{array}$

b)

To determine

To draw: a graph showing feasible region and its coordinates.

Explanation of Solution

Given Information

Let the numbers of sheds be ‘ x’ and the number of play houses be ‘ y’ .

Time required to paint 1 shed = 2.5 days and the required to paint 1 playhouse = 2 days

But, Sean has a total of 20 days to paint

Now we will plot all these inequality in graph

  

Co-ordinates in the feasible region:

A = (0, 40)

B = (45, 0)

P = (25, 20)

c)

To determine

To find: how many sheds and play houses must be painted

Answer to Problem 25PPS

(c) 25 sheds and 20 play houses

Explanation of Solution

Given Information

Let the numbers of sheds be ‘ x’ and the number of play houses be ‘ y’ .

Time required to paint 1 shed = 2.5 days and the required to paint 1 playhouse = 2 days

But, Sean has a total of 20 days to paint

 

Sean should paint 25 sheds and 20 play houses, in order to maximize the profit.

d)

To determine

To find: how many sheds and play houses must be painted for maximum profit

Answer to Problem 25PPS

(d) So, maximum profit= $1250

Explanation of Solution

Given Information

Let the numbers of sheds be ‘ x’ and the number of play houses be ‘ y’ .

Time required to paint 1 shed = 2.5 days and the required to paint 1 playhouse = 2 days

But, Sean has a total of 20 days to paint

Maximum profit

Profit per shed = $ 26

Profit per play house = $ 30

So, maximum profit

  $=(\$26\times 25)+(\$30\times 20)$

  $\begin{array}{l}=\$650+\$600\\ =\$1250\end{array}$