Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.

Question

A computer store owner estimates that by charging dollars each for a certain computer, he can sell computers each week. The quadratic equation   is used to find the revenue, , received when the selling price of a computer is .

Answer 1

Background

When dealing with quadratic equations, if you see the word "maximum" or "minimum", you should think "vertex". When graphed, a quadratic equation takes the shape of a parabola - either one that looks roughly like a smile, or roughly like a frown. For those that look like a smile, the vertex is the lowest point on the graph, and is therefore a minimum; for those that look like a frown, the vertex is the highest point on the graph, and is therefore a maximum.

Solving the problem

The problem is asking us to find the maximum for a given quadratic equation, meaning we need to find its vertex. This will be a single (x, R) point on the graph. When quadratic equations are in the form ax² + bx + c, which this one is, we have an equation for finding the x-coordinate of the vertex:

x = -b / 2a

In the equation R = -x² + 60x, we can see that a is equal to -1 and b is equal to 60 (the coefficients of the x² and x terms, respectively). We can then calculate the x-coordinate of the vertex:

x = -60 / (2 * -1)

x = -60 / -2

x = 30

So when the quadratic reaches its maximum value, x is 30. This means that the selling price to give the maximum revenue is $30.

To find the R-coordinate of the vertex, we plug the x-coordinate back into our quadratic equation:

R = -x² + 60x

R = -(30²) + (60*30)

R = -900 + 1800

R = 900

So when the quadratic reaches its maximum value, R is 900. This means that the maximum revenue is $1800.