# Math Help - Equilibrium Revenue

1. ## Equilibrium Revenue

Please Help. I have a midterm, and this is an example question of what might be on it.

The manager of a CD store has found that if the price of CD is p(x)=80-x/6, then x CDs will be sold. Find an expression for the total revenue from the sale of x CDs (hint: revenue= demand x price). Use your expression to determine the maximum revenue.

2. Revenue is price*demand, so the price(x) here is 80-x/6 and the demand is x, so the expression would be x*(80-x/6)= 80x - x^2/6. Max revenue, in terms of microeconomics, is usually at equilibrium or the midpoint on the demand curve. If thats given, then you just plug it into the expression

3. Originally Posted by aphan19
Please Help. I have a midterm, and this is an example question of what might be on it.

The manager of a CD store has found that if the price of CD is p(x)=80-x/6, then x CDs will be sold. Find an expression for the total revenue from the sale of x CDs (hint: revenue= demand x price). Use your expression to determine the maximum revenue.
is the formula: $p(x) = \frac {80 - x}6$ ? if so, you should use parentheses to indicate that.

Revenue = demand * Price, that is, it is the number of items sold times the price each item is sold for.

thus the revenue function is given by:

$R(x) = x \cdot \frac {80 - x}6 = \frac {40}3x - \frac 16 x^2$

this is a parabola, it's maximum occurs at its vertex

the vertex of a parabola, $f(x) = ax^2 + bx + c$ is the point $\left( \frac {-b}{2a}, f \left( \frac {-b}{2a}\right)\right)$

the answer to your last question is the y-coordinate of that formula

can you find it?

(you could also complete the square to get the vertex, which method do you prefer?)

4. It is not (80-x)/6, it is actually 80-(x/6). Sorry, for not adding the parenthesis in my problem before. If the problem is actually 80-(x/6), now what do I do to solve it?

5. Originally Posted by aphan19
It is not (80-x)/6, it is actually 80-(x/6). Sorry, for not adding the parenthesis in my problem before. If the problem is actually 80-(x/6), now what do I do to solve it?
do exactly what we said before: multiply through by x, and solve for the vertex of the resulting parabola by whatever method you feel comfortable with