SOLUTION: Find the price that will maximize profit for the demand and cost functions, where p is the price, x is the number of units, and C is the cost. Demand Function Cost Function

Algebra ->  College  -> Linear Algebra -> SOLUTION: Find the price that will maximize profit for the demand and cost functions, where p is the price, x is the number of units, and C is the cost. Demand Function Cost Function       Log On


   

Question 1204775: Find the price that will maximize profit for the demand and cost functions, where p is the price, x is the number of units, and C is the cost.
Demand Function Cost Function
p = 80 − 0.1sqrtx C = 35x + 600
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

Demand Function
p%28x%29+=+80+-+0.1sqrt%28x%29+
Cost Function
C%28x%29+=+35x+%2B+600

To find the price that will maximize profit, using demand and cost functions, first step is to find the profit function P%28x%29. Then, use the first derivative to find the critical points for the profit function. Finally use the critical points to find the price to maximize profit.

Profit Function:
P%28x%29+%29+=+R%28x%29+-+C%28x%29+...revenue R%28x%29+=x%2Ap%28x%29
P%28x%29+%29+=+x%2Ap%28x%29+-+C%28x%29+
+P%28x%29+%29+=+x%2A%2880+-+0.1sqrt%28x%29+%29+-+%2835x+%2B+600%29

P%28x%29+=+80x+-+0.1x%5E%283%2F2%29+-+35x+-600+

Then find first derivate and set equal to zero:

%28d%2Fdx%29%2880x+-+0.1x%5E%283%2F2%29+-+35x+-600+%29=-0.15+%28sqrt%28x%29+-+300%29
-0.15+%28sqrt%28x%29+-+300%29=0
sqrt%28x%29+-+300=0
sqrt%28x%29+=300
x=300%5E2
x+=+90000

The maximum profit is P%2890000%29'.