Profit The profit P in thousands of dollars that a manufactu

Profit The profit P, in thousands of dollars, that a manufacturer makes is a function of the number N of items produced in a year, and the formula is P = -0.2N^2 + 3.6N - 9. Express using functional notation the profit at a production level of 5 items per year, and then calculate that value. Determine the two break-even points for this manufacturer-that is, the two production levels at which the profit is zero. Determine the maximum profit if the manufacturer can produce at most 20 items in a year.

Solution

a)
P = - 0.2*N^2 + 3.6*N - 9
at N=5
P = -0.2*5^2 + 3.6*5 - 9
    = - 5+18-9
    = 4

b)
P = - 0.2*N^2 + 3.6*N - 9
- 0.2*N^2 + 3.6*N - 9 = 0
N = {-b +/- sqrt (b^2-4ac) } /2a
     = {-3.6 +/- sqrt (3.6^2 -4*(-0.2)*(-9))} / (2*-0.2)
     = {-3.6 +/- 2.4} / (-0.4)
N =3 or N = 15

c)
dP/dN = -0.4*N + 3.6
equate dP/dN = 0
-0.4*N + 3.6 = 0
N = 9

N=9 maximises profit

Pmax =   - 0.2*N^2 + 3.6*N - 9
             = - 0.2*9^2 + 3.6*9 - 9
            = 1.8