Maximum Profit: Need help with B and C Please The total cost C(q) of producing q
ID: 2831173 • Letter: M
Question
Maximum Profit: Need help with B and C Please
The total cost C(q) of producing q goods is given by: C(q) = 0.01q3 - 0.6q2 + 13q. What is the fixed cost? The fixed cost is $ 0 What is the maximum profit if each item is sold for S11 ? (Assume you sell everything you produce.) Round our answer to two decimal places. The maximum profit is $ Suppose exactly 38 goods are produced. They all sell when he price is $11 each, but for each S1 increase in price, 2 fewer goods are sold. Should the price be increased, and if o by how much? Enter 0 if the price should not be increased. In order to maximize the profit, the price should be increased by $Explanation / Answer
Total revenue of selling q items = 11*q
Profit P = Revenue - Cost
= 11*q - (0.01*q^3 - 0.6*q^2 + 13q)
= -0.01*q^3 + 0.6*q^2 - 2q
For max. profit, dP / dq = 0
dP/dq = -0.01*3*q^2 + 0.6*2q - 2
= -0.03*q^2 + 1.2*q - 2
Setting ti to zero we get, -0.03*q^2 + 1.2*q - 2 = 0
Solving, q = 1.74, 38.25.
Rounding them we get q = 2, 38
For q = 2 we get Profit P = -0.01*2^3 + 0.6*2^2 - 2*2 = -1.68
For q = 38, we get Profit P = -0.01*38^3 + 0.6*38^2 - 2*38 = 241.68
Hence, max. profit = 241.68
c)
When selling price is 11 + 1*x, number of goods sold = 38 - 2*x
Revenue = (11+x)(38-2x) = -2*x^2 + 16x + 418
Cost of producing (38-2x) goods is 0.01*(38-2x)^3 - 0.6*(38-2x)^2 + 13*(38-2x)
Profit P = revenue - Cost
= (-2*x^2 + 16x + 418) - (0.01*(38-2x)^3 - 0.6*(38-2x)^2 + 13*(38-2x))
= 0.08 x^3- 4.16 x^2+ 37.44 x + 241.68
dP/dx = 0 gives
0.24*x^2 - 8.32*x + 37.44 = 0
or x = 5.3, 29.4
When x = 5.3 we get number of goods sold = 38-2*5.3 = 27.4 or 27
When x = 29.4 we get number of goods sold = 38 - 2*29.4 = -20.8
Hence, we take number of goods sold = 27.
Their price = 11 + 5.3 = 16.3
Cost of producing 27 items = 0.01*27^3 - 0.6*27^2 + 13*27 = 110.43
Revenue = 16.3*27 = 440.1
Profit = 440.1 - 110.43 = 329.67
This is higher than profit found in part b.
Hence, he should increae the price to 16.3