Question

For a certain company, the cost for producing x items is 55x+300 and the revenue for selling x items is 95x−0.5x2.

Part a: Set up an expression for the profit from producing and selling x items. We assume that the company sells all of the items that it produces. (Hint: it is a quadratic polynomial.)
Part b: Find two values of x that will create a profit of $300.
Part c: Is it possible for the company to make a profit of $15,000?

Answer 1

Explanation:

Part a: The profit from producing and selling x items can be calculated by subtracting the cost of producing x items from the revenue earned from selling x items. Thus, the profit function P(x) can be expressed as:

P(x) = (95x - 0.5x^2) - (55x + 300)

P(x) = 39.5x - 0.5x^2 - 300

Therefore, the profit function is a quadratic polynomial with a coefficient of -0.5 for the x^2 term.

Part b: To find two values of x that will create a profit of $300, we can set the profit function equal to 300 and solve for x. Thus,

300 = 39.5x - 0.5x^2 - 300

0.5x^2 - 39.5x + 600 = 0

Solving this quadratic equation gives us:

x = 12 or x = 27

Therefore, the company will make a profit of $300 if it produces and sells 12 or 27 items.

Part c: To determine if it is possible for the company to make a profit of $15,000, we can set the profit function equal to 15,000 and solve for x. Thus,

15,000 = 39.5x - 0.5x^2 - 300

0.5x^2 - 39.5x + 15,300 = 0

Solving this quadratic equation gives us:

x = 80 or x = 310

However, since the maximum number of items the company can produce is limited by its resources, the solution x = 310 is not realistic. Therefore, the company cannot make a profit of $15,000.