Question

The monthly demand function for a product sold by a monopoly is p = 1,936 - (1/3)x^2 dollars, and the average cost is C = 1,000 + 10x + x^2 dollars. Production is limited to 1,000 units, and x is in hundreds of units. Find the revenue function, R(x). R(x) = 1936x - (1/3)x^3 Find the cost function, C(x). C(x) = 1,000 + 10x + x^2 Find the profit function, P(x). P(x) = R(x) - C(x) (a) Find P'(x). P'(x) = dP(x)/dx Considering the limitations of production, find the quantity (in hundreds of units) that will give the maximum profit. hundred units (b) Find the maximum profit. (Round your answer to the nearest cent.) $

          The monthly demand function for a product sold by a monopoly is p = 1,936 - (1/3)x^2 dollars, and the average cost is C = 1,000 + 10x + x^2 dollars. Production is limited to 1,000 units, and x is in hundreds of units.

Find the revenue function, R(x).
R(x) = 1936x - (1/3)x^3

Find the cost function, C(x).
C(x) = 1,000 + 10x + x^2

Find the profit function, P(x).
P(x) = R(x) - C(x)

(a) Find P'(x).
P'(x) = dP(x)/dx

Considering the limitations of production, find the quantity (in hundreds of units) that will give the maximum profit.
hundred units

(b) Find the maximum profit. (Round your answer to the nearest cent.)
$

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Added by Evan T.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Eduard Sanchez

04/09/2023

Step 1

The revenue function is the product of the price function and the quantity, so: R(x) = p(x) * x = (1936 - \frac{1}{3}x^2) * x = 1936x - \frac{1}{3}x^3 Show more…

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The monthly demand function for a product sold by a monopoly is p = 1,936 - (1/3)x^2 dollars, and the average cost is C = 1,000 + 10x + x^2 dollars. Production is limited to 1,000 units, and x is in hundreds of units. Find the revenue function, R(x). R(x) = 1936x - (1/3)x^3 Find the cost function, C(x). C(x) = 1,000 + 10x + x^2 Find the profit function, P(x). P(x) = R(x) - C(x) (a) Find P'(x). P'(x) = dP(x)/dx Considering the limitations of production, find the quantity (in hundreds of units) that will give the maximum profit. hundred units (b) Find the maximum profit. (Round your answer to the nearest cent.) $

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Transcript

00:01 Hi there.

00:02 So for this program, we are told that the monthly demand function for a project sold by a monopoly is given by the following expression.

00:09 Then the price for this is equal to one thousand nine hundred and thirty six is minus one divided by three times x.

00:20 Oh, sorry.

00:20 It's one three times x as four and the average cost for this.

00:27 So we are given the average cost expression that is one thousand plus ten times x plus x squared.

00:35 Now, with that said, remember, the production is limited to one thousand units or which is the same ten hundred units.

00:45 Ok, so we need to find the revenue function first.

00:50 So the revenue function, remember that that is just equal to.

00:58 Ok, the revenue is just the price times x.

01:01 So for this case, we just multiply the price that we're given by x.

01:06 So that will give us one thousand nine hundred and thirty six.

01:10 This times x minus one divided by three times x to the three.

01:14 So that is the expression for the revenue.

01:17 The first question for this.

01:19 Now, next question for this, and we are asked to find the cost function.

01:23 Now, remember that what we are given is the average cost.

01:26 And we know that the average cost is just the cost function divided by x.

01:31 So since we want the cost, we just need to multiply the average cost that we are given by x.

01:38 So when we do that, we obtain the following.

01:41 We obtain one thousand times x plus ten times x squared plus x to the three.

01:49 So that is the expression for the cost.

01:52 Now, let's keep going.

01:54 Ok, the next question is to find the profit.

01:57 So the profit, we know that that is the revenue minus the cost...

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