# Find the number of units to produce at each plant in order to maximize the total profit and find the maximum total profit...A company has an item produced in plants A and B. The cost of producing x...

Find the number of units to produce at each plant in order to maximize the total profit and find the maximum total profit...

A company has an item produced in plants A and B. The cost of producing x units at plant A is C1=0.02x^2+5x+1000 and the cost of producing y units at plant B is C2=0.05y^2+y+100. All units produced sell for $ 15. The total profit function is P(x,y)=15(x+y)-C1-C2.

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### 1 Answer

The company has an item produced in plants A and B. The cost of producing x units at plant A is C1=0.02x^2+5x+1000 and the cost of producing y units at plant B is C2=0.05y^2+y+100. All units produced sell for $ 15. The total profit function is P(x,y)=15(x+y)-C1-C2

The profit made when x units are produced in plant A is P1 = 15*x - 0.02x^2 - 5x - 1000

To maximize P1, solve P1' = 0

=> 15 - 0.04x - 5 = 0

=> 0.04*x = 10

=> x = 250

The profit made when y units are produced in plant B is P2 = 15*y - 0.05y^2 - y - 100

To maximize P2, solve P2' = 0

=> 15 - 0.1y - 1 = 0

=> 0.1*y = 14

=> y = 140

The total profit is 15(x+y) - C1 - C2

The maximum total profit is 15*390 - 3500 - 1220 = 1130

**The maximum profit is $1130**