Donuts Delights, Inc. Has determined that when x donutsare made daily, the profit P (in dollars) is given by P (x) = - 0.001 x2+3.2x-1450 (a) What is the company's profit if 700 donutsare made daily? (b) How many donuts should be made daily in order to maximize the company's profit?

Question

Mathematics

Pace

20 August, 04:42

Donuts Delights, Inc. Has determined that when x donutsare made daily, the profit P (in dollars) is given by P (x) = - 0.001 x2+3.2x-1450 (a) What is the company's profit if 700 donutsare made daily? (b) How many donuts should be made daily in order to maximize the company's profit?

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Jair Johnson · Answer 1

Step-by-step explanation:

In order to find the profit made by making 700 donuts daily, simply evaluate the polynomial (quadratic, to be more specific) at an x value of 700:

P (700) = -.001x² + 3.2x - 1450 and

P (700) = $300

Now, keeping in mind this is in fact a quadratic, it has a vertex. Because this is a negative parabola (the shape created by a quadratic function), the vertex indicates the max value of the function. By putting this into vertex form by completing the square, we can determine the vertex. From the vertex, we can determine not only the number of donuts that will maximize the profit, but the profit as well.

To complete the square, first set the quadratic equal to 0, then move the constant over to the other side of the equals sign:

-.001x² + 3.2x = 1450

The rule for completing the square is that the leading coefficient has to be a positive 1. Right now ours is a negative. 001, so we have to factor it out:

-.001 (x² - 3200x) = 1450

Now the rule is to take half the linear term, square it, and add it to both sides of the function. Our linear term is 3200. Half of that is 1600; 1600 squared is 2560000. BUT we cannot ignore the -.001 out front of the parenthesis. It is a multiplier. That means that we didn't just add 2560000 to the left, we added -.001 (2560000) which is - 2560:

-.001 (x² - 3200x + 2560000) = 1450 - 2560

The reason we did this is to create a perfect square binomial on the left. We will state that binomial and at the same time, simplify the right:

-.001 (x - 1600) ² = - 1110

Now we can put everything back on the left and set it back equal to y:

-.001 (x - 1600) ² + 1110 = y

From this we can ascertain that the vertex is a max value located at

(1600, 1110).

In this situation, the 1600 represents the number of donuts that will maximize the profit, and the profit is $1110.