Two Important and similar Ideas to the marginal cost are the marginal revenue an
ID: 2876791 • Letter: T
Question
Two Important and similar Ideas to the marginal cost are the marginal revenue and profit from producing a certain number of Items. Consider the cost function C(x) = 900 - 4x + 0.1x^2 dollars, which gives the cost C to a company of producing x books. In general, the more items one produces, the less revenue one obtains per item from their sale. Suppose that when we produce x books, we can sell them for 52 - 0.1x dollars each. (a) What is the revenue R(x) from selling x books? R(x) = (b) What is the profit P(x) from selling x books? (Consider the revenue and cost functions.) p(x) = (c) What Is the actual additional revenue from selling the 101st book? (Round your answer to the nearest cent.) What is the additional profit? (Round your answer to the nearest cent.) $ (d) What is the rate T"(x) at which the revenue is changing when 100 books are being produced? Include units. How does this compare with part (c)? This is a cents difference from the additional revenue found in part (c). Is the revenue increasing or decreasing with respect to x? increasing decreasing (e) What is the rate P'(x) at which the profit is changing when 100 books are being produced? Include units. How does this compare with part (c)? This is a cents difference from the additional profit found in part (c). (e) What is the rate P'(x) at which the profit is changing when 100 books are being produced? Include units. How does this compare with part (c)? This is a cents difference from the additional profit found in part (c). Is the profit increasing or decreasing with respect to x? increasing decreasing (f) For what x would the revenue be a maximum? x = What is the maximum revenue? $ (g) For what x would the profit be a maximum? x = What is the maximum profit? $Explanation / Answer
C(x) = 900 - 4x + 0.1x^2
p = 52 - 0.1x
(a) R(x) = px = 52x - 0.1x^2
(b) P(x) = R(x) - C(x) = 52x - 0.1x^2 - (900 - 4x + 0.1x^2) = -0.2x^2 + 56x - 900
(c) R(101) = 52(101) - 0.1(101)^2 = 4231.90
R(100) = 52(100) - 0.1(100)^2 = 4200
Additional revenue = 4231.90 - 4200 = 31.90
P(101) = -0.2(101)^2 + 56(101) - 900 = 2715.80
P(100) = -0.2(100)^2 + 56(100) - 900 = 2700
Additional profit = 2715.80 - 2700 = 15.80
(d) R'(x) = 52 - 0.2x
R'(100) = 52 - 0.2(100) = 32
This is a 10 cents difference (10 cents more) from the additional revenue found in part (c)
The revenue is increasing with respect to x
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