Break-Even Points Suppose that the cable television company’s cost function in Example cha...

Break-Even Points Suppose that the cable television company’s cost function in Example changes to C(x) = 275 + 12x. Determine the new break-even points.

Example

Profit and Break-Even Points A cable television company estimates that with x thousand subscribers its monthly revenue and cost (in thousands of dollars) are

R(x) = 32x − .21x2

C(x) = 195 + 12x.

Determine the company’s break-even points; that is, find the number of subscribers at which the revenue equals the cost. (See Fig. 1.)

Figure 1 Break-even points.

SOLUTION

Let P(x) be the profit function:

P(x) = R(x) − C(x)

= (32x − .21x2 ) − (195 + 12x)

= −.21x2 + 20x − 195.

The break-even points occur where the profit is zero. Thus, we must solve

−.21x2 + 20x − 195 = 0.

From the quadratic formula,

The break-even points occur where the company has 11,030 or 84,210 subscribers. Between those two levels, the company will be profitable.

Factoring If f(x) is a polynomial, we can often write f(x) as a product of linear factors (i.e., factors of the form ax + b). If this can be done, then, we can determine the zeros of f(x) by setting each linear factor equal to zero and solving for x. (The reason is that the product of numbers can be zero only when one of the numbers is zero.)