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March 10th, 2013, 10:05 AM   #1
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Total Revenue as a function of Quantity

Hi,

I'm new here and a little dim so please be gentle.

I'm having trouble with understanding total revenue in terms of quantity. The example is below with my questions in bold.

Any help would be greatly appreciated.

Here's the example:

Given the demand function
P = 100 − 2Q
express TR as a function of Q and hence sketch its graph.
(a) For what values of Q is TR zero?
(b) What is the maximum value of TR?

Solution
Total revenue is defined by TR = PQ and, since P = 100 − 2Q, we have
TR = (100 − 2Q)Q = 100Q − 2Q²

How was this turned into a quadratic equation and why multiply the whole function by Q?

This function is quadratic and so its graph can be sketched
Step 1
The coefficient of Q² is negative, so the graph has an inverted U shape.
Step 2
The constant term is zero, so the graph crosses the TR axis at the origin. (which constant term is 0, is it because the equation does not have a c as in x = ax2 + bx + c
Step 3
To find where the curve crosses the horizontal axis, we could use "the formula". However, this is not necessary,
since it follows immediately from the factorization
TR = (100 − 2Q)Q
that TR = 0 when either 100 − 2Q = 0 or Q = 0. In other words, the quadratic equation has two solutions,
Q = 0 and Q = 50.
The total revenue curve is shown in Figure 2.6.
From Figure 2.6 the total revenue is zero when Q = 0 and Q = 50.
By symmetry, the parabola reaches its maximum halfway between 0 and 50, that is at Q = 25. The
corresponding total revenue is given by
TR = 100(25) − 2(25)² = 1250
Practice Problem
1 Given the demand function
P =

Last edited by skipjack; November 19th, 2014 at 06:17 AM.
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March 11th, 2013, 10:56 AM   #2
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Re: Total Revenue as a function of Quantity

Quote:
Originally Posted by Azhar
Given the demand function
P = 100 − 2Q
express TR as a function of Q and hence sketch its graph.
(a) For what values of Q is TR zero?
(b) What is the maximum value of TR?

Solution
Total revenue is defined by TR = PQ and, since P = 100 − 2Q, we have
TR = (100 − 2Q)Q = 100Q − 2Q²

How was this turned into a quadratic equation and why multiply the whole function by Q?
Total revenue is price, 100 - 2Q, times quantity Q. So (100 - 2Q) * Q = 100Q - 2Q^2.

Last edited by skipjack; November 19th, 2014 at 06:19 AM.
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