Hi.

My name Devon and I am a high school student tutoring grade 9 math. I am having problem with a question.

The profit a manufacturing company makes can be found using the formula
P=120n - n^2 - 2200

How many items must the company produce to begin to make a profit? The answer is 23 in the answer key but I cant figure out the procedure What part of math is this sort of problem? algebra? factoring?

Thank you

Hi Devon. This is indeed an algebra question.

The profit is P and you want the profit to be positive, so P > 0.

Then, since P = 120n - n² - 2200,

120n - n² - 2200 > 0

There are a couple of ways to solve it from here.

The way I would do this is to recognize that this is a quadratic expression (because there is just one variable, n, and its largest exponent is a 2). So that means it is a parabola. If I look at the sign in front of the n² term, it is a minus, so that means it is a parabola that opens down.

Now the Quadratic Equation can tell us what values of n will solve this: -n² + 120n - 2200 = 0. The Quadratic equation will give you two values for n in this case, both of them positive values. That means that the parabola that opens down crosses the x-axis twice (the quadratic formula tells you where a parabola crosses the axis). That means the zone between this points is a positive value of P.

Now since this is a manufacturing problem, you are dealing with a whole number of units. That means n must be the smallest positive whole number that is between the two values that the quadratic equation gave you.

It is an interesting problem, because for this particular question, the company makes profit only when it develops a certain quantity of the products. If it produces too many, it will also lose money (P is negative). Can you tell what is the maximum number of units the company can produce and still make a profit?

Hope this helps you,
Stephen La Rocque.