Question

The financial department of a company that produces memory chips for microcomputers arrived at the following price-demand function and the corresponding revenue function:

p(x) = 75 – 3x price-demand

R(x) = x ? p(x) = x(75 – 3x) revenue function

Where p(x) is the wholesale price in dollars at which x million chips can be sold and R(x) is in millions of dollars. Both functions have domain 0 ? x ? 20. They also found the cost function to be C(x) = 125 + 16x for manufacturing and selling x million chips. Find the profit function and determine the number of chips that should be sold for maximum profit. Find the maximum profit. Find the maximum revenue and the number of chips that need to be sold to reach that maximum. Graph the cost, revenue, and profit functions in the same window on your calculator. (Use window x = 0 to 20, and y = 0 to 600). Round your answers to the nearest whole number. Round the numbers of chips up to the next whole number.

a.) Profit function is simplified form __________

b.) Number of chips for maximum profit _________

c.) Number of chips for maximum revenue _______

d.) Maximum profit ______

e.) Maximum revenue _______

f.) What is the profit when the revenue is at its maximum? _______

g.) Explain why the maximum revenue is different from the maximum profits_____________

**Please show ALL work!

Answer 1

(a)

Profit = Revenue - Cost = R(x) - C(x)

(b)

To find the number of chips for maximum profit, we find the first derivative of profit function with respect to x and put it equal to zero.

After rounding off, number of chips for maximum profit are 10

(c)

Revenue = R(x) = x(75 – 3x) = 75x - 3x²

To maximize revenue, we find the first derivative of R(x) with respect to x and put it equal to zero.

After rounding off, number of chips for maximum revenue are 13

(d)

Maximum profit will be found by putting x=10 in profit function

So, maximum profit is 165

(e)

Maximum profit will be found by putting x=13 in revenue function

R(x) = 75x - 3x²

R(13) = 75(13)- 3(13)² =975 - 507 = 468

So, maximum revenue is 468

(f)

Profit when revenue is maximum will be found by putting x = 13 in profit function

So, profit when revenue is maximum will be 135

(g)

Maximum revenue is different from maximum profit because cost is not considered in maximizing revenue function. When this happens, the producer will overproduce the quantity of good by 3 units (13-10) and hence reduce the amount of profit since it is not optimal quantity.