Astore sells two types of copy machines: compact (low capacity) and standard (which takes more space). the store can sell up to 90 copiers a month. a maximum of 1080 cubic feet of storage space is available. a compact copier requires 6 cu. ft. of storage space, and a standard copier requires 18 cu. ft. the compact and standard copy machines take, respectively, 1 and 1.5 sales hours of labor. a maximum of 99 hours of labor is available. the profit from each of these copiers is $60 and $80 respectively. what is the maximum profit? round your answer to the nearest whole dollar. do not include a dollar sign or comma in your answer.

$5,760

Step-by-step explanation:

This is a problem of linear programming on two variables which can be solved graphically.

Let x ≥ 0 and y ≥ 0 the number of units of compact and standard machines respectively.

Restrictions

“The store can sell up to 90 copiers a month”

x+y ≤ 90

(See picture 1 attached)

“A compact copier requires 6 cu. ft. of storage space, and a standard copier requires 18 cu. ft. and maximum of 1080 cubic feet of storage space is available.”

6x + 18y ≤  1080

(See picture 2 attached)

“The compact and standard copy machines take, respectively, 1 and 1.5 sales hours of labor. A maximum of 99 hours of labor is available.”

1x + 1.5y ≤ 99

(See picture 3 attached)

Vertex V1 = (18, 54) is the intersection of the lines

6x +18y = 1080

x + 1.5y = 99

which is the solution of the linear system of equations.

Vertex V2 = (72, 18) is the intersection of the lines

x + y = 90

x + 1.5y = 99

We want to find the maximum profit, i.e., the maximum of  

P = 60x +80y

on the region obtained by intersecting all the above regions.

Since P is a linear function, the maximum is attained on one of the vertexes  

(0, 60), (18, 54), (72, 18), (90, 0)

Let's evaluate P at each of this points to find the maximum out.

P(0, 60) = 80*60 = $480

P(18, 54) = 60*18 + 80*54 = $5,400

P(72, 18) = 60*72 + 80*18 = $5,760

P(90, 0) = 60*90 = $5,400

And we can see that the maximum is obtained when 72 units of compact machines and 18 units of standard machines are sold, and the maximum profit is $5,760.