Economics

Marginal revenue
Suppose that...

Question

Economics

Marginal revenue

Suppose that the revenue from selling x washing machines is

r(x) = 20000(1 − 1/x) dollars.

b. Use the function r'(x) to estimate the increase in revenue that will result from increasing production from 100 machines a week to 101 machines a week.

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A company produces 10,000 paper cups per day. And their revenue from selling these is given by the function R of X is equal to 5000 multiplied by 1 minus 2 divided by X dollars. Estimate the increase in revenue from increasing production from 50,000 to 51,000 paper cups per day. Awesome. So it appears for this particular prompt, we're asked to ultimately, we're trying to figure out what the we need to estimate our increase in the revenue when we increase our production from 50,000 to 51,000 paper cups per day. So we're ultimately trying to figure out this estimate in the increase in revenue. So that's our value that we're ultimately trying to solve for. So with that said, let us read off our multiple choice answers to see what our final answer might be. A is $3.25 B is $3.50 C is $4 and D is $4.50. Fantastic. So first off, let us note and recall that we can estimate the increase in revenue using the derivative of the revenue function, so meaning if we take R of X is equal to 5000 multiplied by 1 minus 2 divided by X. And we take the derivative, so we take R of X, where this prime symbol means the first derivative of R of X. So if we take R of X is equal to 5000 multiplied by D by DX of 1 minus 2 divided by X, we will find that R of X is equal to 5000 multiplied by D by DX. Of 1 minus 2 X to the power of -1. So that will mean that R of X is equal to 5000 multiplied by 0 + 2 X to the power of -2. So that will mean that R X when we perform the derivative, R X will be equal to. 10,000 divided by X to the power of 2. So now we need to evaluate our derivative at X equals 50. So when we do that, we will find that R of 50 is going to be equal to 10,000 divided by 50 to the power of 2, and when we plug that into our calculator, we will find that it's equal to 4. So since we're increasing the production by 1000 cups, which means from 50 to 51,000 paper cups per day, the derivative that we just found together will give us an estimate of the increase in revenue, which will be approximately. $4 and that's it. We've officially solved for this problem. So therefore, our estimation in the increase of revenue is going to be approximately $4. And that will mean that our final answer has to be, without a doubt, multiple choice answer, letter C, which once again is $4. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Economics

Marginal revenue
Suppose that the revenue from selling x washing machines is

r(x) = 20000(1 − 1/x) dollars.

b. Use the function r'(x) to estimate the increase in revenue that will result from increasing production from 100 machines a week to 101 machines a week.

Key Concepts

Derivative

Marginal Revenue

Function Evaluation

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