Question

A market analyst working for a small-appliance manufacturer finds that the firm produces and sells x blenders annually, the total profit (in dollars) is P(x) = 82 + 0.33x - 0.00133x^2.

a. How many blenders must the firm produce to break even?

Answer 1

To find the number of blenders the firm must produce to break even, set the profit function equal to zero and solve for x using the quadratic formula. The break even point is the positive root of the equation obtained.

Step-by-step explanation:

The student is asking about a function that represents the profit from producing and selling blenders annually. To break even, the profit must be zero, which is when total revenue equals total costs. By setting the profit function P(x) =
`82 + 0.33x - 0.00133x^(2)` equal to zero and solving for x, we can find the number of blenders that must be produced to break even.

To solve P(x) = 0, we need to find the roots of the equation
`0.00133x^(2) - 0.33x - 82` = 0. This requires using the quadratic formula. After calculation, the firm must produce the number of blenders at the positive root of this equation to break even, as negative production is not feasible.