Homework SourseA manufacturer produces 4 Mb flash drives at 2 per unit She
A manufacturer produces 4 Mb flash drives at $2 per unit. She sells them for $5 apiece, and at that price consumers have been buying them at 4,000 units a month. The manufacturer plans to raise the price; market research indicates that for every $1 increase in price, 400 fewer units will be sold each month.
(A) Find the revenue function, R, as a function of unit price, p.
Find the cost, C, as a function of p.
Find the profit, P, as a function of p.
(B) How many units must the company sell to break even?
(C) How much per unit must the company sell to attain the maximum profit?
[Do not use derivatives in this problem.]
(D) What is the maximum profit?
(E) How many units are sold at this price?
(A) Find the revenue function, R, as a function of unit price, p.
Find the cost, C, as a function of p.
Find the profit, P, as a function of p.
(B) How many units must the company sell to break even?
(C) How much per unit must the company sell to attain the maximum profit?
[Do not use derivatives in this problem.]
(D) What is the maximum profit?
(E) How many units are sold at this price?
Solution
(A) No. of units sold per month as a function of p is = 4000 - (p-5)*400
Revenue R = p*(4000 - (p-5)*400)
Cost C = 2*(4000 - (p-5)*400)
Profit P = R - C = (4000 - (p-5)*400)*(p-2) = 6800p - 12000 - 400p^2
(B) For breakeven, P = 0.
0 = 6800p - 12000 - 400p^2 = 400(17p - 30 - p^2)
p = 15, 2
For p = 0 to 15, P is negative. Hence, break even is at p =15.
No. of units at breakeven = 4000 - (15-5)*400 = 0
(C) Graph of P versus p shows that P is max. at p = 8 and 9. Hence selling at $8 or $9 per unit will maximize the profit.
(D) Max. profit = 6800*8 - 12000 - 400*8^2 = $16,800
(E) Units sold = 4000 - (8-5)*400 = 2800
