1 Let Cx 11z7 be the cost in hundreds of dollars to produce

1. Let C(x) 11z7 be the cost (in hundreds of dollars) to produce z units of a certain commodity, and the R(z) =-x2 + 19x be the revenue (in hundreds of dollars) from the sale of x units of that commodity. How many units must the manufacturer of the commodity produce and sell in order to maximize their profit? What is that maximum profit? Note that profit can be computed by subtracting cost from revenue.

Solution

R = -x^2 + 19x
C = 11x + 7

Profit, P = R - C

P = -x^2 + 8x - 7

Now, for max profit, we basically have to find vertex of this quadratic ...

a = -1 , b = 8

So, x-value of vertex = -b/(2a)
= -8/(2*-1)
= -8/-2
= 4

So, when x = 4, we have
P = -16+ 32 - 7
= 9

So, they must sell 4 units

And the max profit value is 9 hundred dollars, as in 900 dollars