Chapter 2.6, Problem 27E

a.

To determine

To graph: the polygonal convex region represented by the cost of renting space.

Answer to Problem 27E

  

Explanation of Solution

Given information:

Monthly rent of main street site per square foot = $10

Monthly rent of site on high street per square foot = $20

Minimum rent for both the sites per square foot = $20

Potential customers per square foot at main street site = 30

Potential customers per square foot at high street site = 40

Monthly budget for rental space = $1200

Calculation:

Let main street site and high street site be x and y respectively. The inequalities or equation representing the given data will be as follows:

  $\begin{array}{c}x\ge 20\\ y\ge 20\\ 10x+20y=1200\end{array}$

The shaded portion of the graph representing the cost of renting space is as follows:

  

The points satisfying both the inequalities and equation are $\left(20,50\right),\left(20,20\right)$ and $\left(80,20\right)$ .

b.

To determine

To find: the function representing the possible number of customers per square foot at both locations.

Answer to Problem 27E

  $f\left(x,y\right)=30x+40y$

Explanation of Solution

Given information:

Monthly rent of main street site per square foot = $10

Monthly rent of site on high street per square foot = $20

Minimum rent for both the sites per square foot = $20

Potential customers per square foot at main street site = 30

Potential customers per square foot at high street site = 40

Monthly budget for rental space = $1200

Calculation:

The inequalities derived from the given data are as follows:

  $\begin{array}{c}x\ge 20\\ y\ge 20\\ 10x+20y=1200\end{array}$

Here main street site and high street site be x and y respectively.

Thus, the function representing the possible number of customers per square foot at both locations is as follows:

  $f\left(x,y\right)=30x+40y$

c.

To determine

To find: the space (in square feet) that should be rented at each site to maximize the number of potential customers.

Answer to Problem 27E

3200

Explanation of Solution

Given information:

Monthly rent of main street site per square foot = $10

Monthly rent of site on high street per square foot = $20

Minimum rent for both the sites per square foot = $20

Potential customers per square foot at main street site = 30

Potential customers per square foot at high street site = 40

Monthly budget for rental space = $1200

Calculation:

The function representing the possible number of customers per square foot at both locations is as follows:

  $f\left(x,y\right)=30x+40y$

Here main street site and high street site be x and y respectively.

From the graph, the points satisfying both the inequalities and equation are $\left(20,50\right),\left(20,20\right)$ and $\left(80,20\right)$ . Put the points in the function $f\left(x,y\right)$ to check the maximum space (in square feet).

  $\begin{array}{c}f\left(20,50\right)=30\left(20\right)+40\left(50\right)\\ =2600\\ \\ f\left(20,20\right)=30\left(20\right)+40\left(20\right)\\ =1400\\ \\ f\left(80,20\right)=30\left(80\right)+40\left(20\right)\\ =3200\end{array}$

Thus, the maximum space (in square feet) to maximize the customers is 3200.

d.

To determine

To decide: and explain reason, whether the space should be rented at one of the sites or both sites by the president.

Answer to Problem 27E

Main Street should be rented because it would have maximum customers.

Explanation of Solution

Given information:

Oil required for one batch of garlic dressing = 2 quarts

Monthly rent of main street site per square foot = $10

Monthly rent of site on high street per square foot = $20

Minimum rent for both the sites per square foot = $20

Potential customers per square foot at main street site = 30

Potential customers per square foot at high street site = 40

Monthly budget for rental space = $1200

Calculation:

The equation for renting to have maximum profit is $10x+20y=1200$ .

If all of the customers went to High Street $\left(x=0\right)$ , there would be a maximum of 60 customers.

If all of the customers went to Main Street $\left(y=0\right)$ , there would be a maximum of 120 customers.

Assuming the president of the space like the most customers, 120 would be the maximum number of customers. Thus, the president will rent Main Street to have maximum customers.