Expert: Jack Cheng - 12/8/2006

Question
A small company that manufactures snowboards uses the relation P=-81X sqaured + 162x to model its prfit. In the model x represents the number of snowbpards in the thousands, and P represents the profit in thousands of dollars.

1)what si the maximum profit the company can earn?
2)how many snowboards must it produce to earn this profit?
3)the company breaks even when there is neither a profit or a loss. What are the break even points for the company?

Answer
Hi Jen,

To find the maximum profit, you take the derivative of the equation and find its x-intercept. That gives you the x for when the company is making the most profit (the graph is parabolic pointing down, that's why the x-intercept of the derivative is the max):

P = -81x^2 + 162x
P' = -162x + 162

0 = -162x + 162
0 = 162(1-x)
1 = x

So when the company creates 1000 snowboards, it's making the most profit. That profit is:

-81(1)^2 + 162(1) = -81 + 162 = 81

or 81,000 dollars.

The company break even when the profit is zero. So, setting P to zero in the equation, we solve for x:

0 = -81x^2 + 162x
0 = 81x(2-x)
x = 0 or 2

or when the company has just produced 0 or 2000 snowboards.

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Another way to solve the first two parts, if you haven't taken calculus, would be to just look at the graph and see where the top of the curve is (it should look like a hill). The y-coordinate of that point is the max profit (times 1000), and the x-coordinate is the number of snowboards the company needs to produce (times 1000).

I hope this helps,
Jack