163- In general, the profit function is the difference between the revenue and c
ID: 2865756 • Letter: 1
Question
163- In general, the profit function is the difference between the revenue and cost functions: P(x) = R(x) C(x). Suppose the price-demand and cost functions for the production of cordless drills is given respectively by p = 143 0.03x and C(x) = 75,000 + 65x, where x is the number of cordless drills that are sold at a price of p dollars per drill and C(x) is the cost of producing x cordless drills.
a. Find the marginal cost function.
b. Find the revenue and marginal revenue functions.
c. Find R(1000) and R(4000). Interpret the results.
d. Find the profit and marginal profit functions.
e. Find P(1000) and P(4000). Interpret the results.
Explanation / Answer
Solution:
Total cost, C(x) = 75000 + 65x
a)
marginal cost, C '(x) = 65
b)
Rvenue function, R(x) = x p(x) = x(143 - 0.03x) = 143x - 0.03x^2
marginal revenue, R '(x) = 143 - 0.06x
c)
marginal revenue, R '(x) = 143 - 0.06x
R'(1000) = 143 - 0.06(1000) = 143 - 60 = 83
R'(4000) = 143 - 0.06(4000) = 143 - 240 = -97 take positive say 97
d)
profit, P(x) = R(x) - C(x)
= 143x - 0.03x^2 - 75000 - 65x
= - 0.03x^2 + 78x - 75000
marginal profit, P '(x) = - 0.06x + 78
(e)
marginal profit, P '(x) = - 0.06x + 78
P'(1000) = -0.06(1000) + 78 = 60 + 78 = 138
P'(4000) = -0.06(4000) + 78 = 240 + 78 = 318
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