Chapter 3.4, Problem 25PPS

a.

To determine

To describe: The system of inequalities that represent the different number of ways he can paint the structures.

Answer to Problem 25PPS

The system of inequalities that represent the different number of ways he can paint the structuresis

  $\begin{array}{c}x\ge 0\\ y\ge 0\\ x+y\le 45\\ 2.5x+2y\le 20\end{array}$

Explanation of Solution

Given information:

He has 45 structures that needs to be painted. Per day cost of painting a shed is 2.5 and per day cost of painting a play house is 2. He has 20 days to paint as many structures as she can.

Calculation:

Consider the provided information thathe has 45 structures that needs to be painted. Per day cost of painting a shed is 2.5 and per day cost of painting a play house is 2. He has 20 days to paint as many structures as she can.

Let x denote the number of sheds painted and y denote the number of play houses painted.

The constraints are,

  $\begin{array}{c}x\ge 0\\ y\ge 0\\ x+y\le 45\\ 2.5x+2y\le 20\end{array}$

b.

To determine

To graph: The region that depicts the represent the different number of ways he can paint the structures.

Explanation of Solution

Given information:

The system of inequalities that represent the different number of trays she can bake is

  $\begin{array}{c}x\ge 0\\ y\ge 0\\ x+y\le 45\\ 2.5x+2y\le 20\end{array}$

Graph:

Consider the provided information system of inequalities that represent the different number of ways he can paint the structures is

  $\begin{array}{c}x\ge 0\\ y\ge 0\\ x+y\le 45\\ 2.5x+2y\le 20\end{array}$

Plot the above inequalities on coordinate plane.

  

Interpretation:

The shaded region represents the feasible. It has 3 corner points. The coordinates of feasible region are $\left(0,0\right),\left(8,0\right)\text{ and }\left(0,10\right)$ .

c.

To determine

To calculate:The number ofeach structure to be painted.

Answer to Problem 25PPS

10 play houses and 0 sheds must be painted to have a profit.

Explanation of Solution

Given information:

He has 45 structures that needs to be painted. Per day cost of painting a shed is 2.5 and per day cost of painting a play house is 2. He has 20 days to paint as many structures as she can.

Calculation:

Consider the provided information thathe has 45 structures that needs to be painted. Per day cost of painting a shed is 2.5 and per day cost of painting a play house is 2. He has 20 days to paint as many structures as she can.

Let x denote the number of sheds painted and y denote the number of play houses painted.

The constraints are,

  $\begin{array}{c}x\ge 0\\ y\ge 0\\ x+y\le 45\\ 2.5x+2y\le 20\end{array}$

Since, he makes a profit of $26per shed and $30 per play house.

The objective function or profit function is,

  $f\left(x,y\right)=26x+30y$

  

The shaded region represents the feasible. It has 3 corner points. The coordinates of feasible region are $\left(0,0\right),\left(8,0\right)\text{ and }\left(0,10\right)$ .

Substitute the vertices of the feasible region to find the point at which maximum profit is there.

Substitute $\left(0,0\right)$ in the function $f\left(x,y\right)=26x+30y$ .

  $f\left(0,0\right)=0$

Substitute $\left(0,10\right)$ in the function $f\left(x,y\right)=26x+30y$ .

  $f\left(0,10\right)=300$

Substitute $\left(8,0\right)$ in the function $f\left(x,y\right)=26x+30y$ .

  $f\left(8,0\right)=208$

Thus, 10 play houses and 0 sheds must be painted to have a profit.

d.

To determine

To calculate:The maximum profit.

Answer to Problem 25PPS

The maximum profit is $\$300$ when 10 play houses and 0 sheds are painted.

Explanation of Solution

Given information:

He has 45 structures that needs to be painted. Per day cost of painting a shed is 2.5 and per day cost of painting a play house is 2. He has 20 days to paint as many structures as she can.

Calculation:

Consider the provided information thathe has 45 structures that needs to be painted. Per day cost of painting a shed is 2.5 and per day cost of painting a play house is 2. He has 20 days to paint as many structures as she can.

Let x denote the number of sheds painted and y denote the number of play houses painted.

The constraints are,

  $\begin{array}{c}x\ge 0\\ y\ge 0\\ x+y\le 45\\ 2.5x+2y\le 20\end{array}$

Since, he makes a profit of $26 per shed and $30 per play house.

The objective function or profit function is,

  $f\left(x,y\right)=26x+30y$

  

The shaded region represents the feasible. It has 3 corner points. The coordinates of feasible region are $\left(0,0\right),\left(8,0\right)\text{ and }\left(0,10\right)$ .

Substitute the vertices of the feasible region to find the point at which maximum profit is there.

Substitute $\left(0,0\right)$ in the function $f\left(x,y\right)=26x+30y$ .

  $f\left(0,0\right)=0$

Substitute $\left(0,10\right)$ in the function $f\left(x,y\right)=26x+30y$ .

  $f\left(0,10\right)=300$

Substitute $\left(8,0\right)$ in the function $f\left(x,y\right)=26x+30y$ .

  $f\left(8,0\right)=208$

Thus, the maximum profit is $\$300$ when 10 play houses and 0 sheds are painted.