The cost function for a company that produces cellular phones is C ( x ) = 11x^2 − 243x + 11682

Question

The cost function for a company that produces cellular phones is 

 C(x)=11x2−243x+11682C(x)=11x2-243x+11682,

where CC is the cost in dollars and xx is the number of cellular phones produced (in thousands).

a. How many units of cellular phones (in thousands) should be produced in order to minimize the cost? Round to the nearest cellular phone unit. ________________(muliplied by 1000) cellphones

b. What is the minimum cost? Round to the nearest cent. __________$  (muliplied by 1000)

Answer 1

The cost equation you've been given is

C(x) = 11x² - 243x + 11682

which is a quadratic function (that is, the variable is to the second power, x²). You should have learned that these functions have a graph that looks like a parabola. Parabolas have a lowest (or highest, if it's inverted) point, called a vertex. Since the problem is asking you to find the lowest possible cost, that means they're asking you to look at the vertex.

There are several ways to look at the vertex, including graphing the function, re-writing it in vertex form, etc. For the purposes of the problem, however, the easiest thing is to remember that in a formula like ax²+bx+c, the x-coordinate of the vertex will be located at -b/(2a). In the case of your formula, the values of a, b, and c are:

a=11, b=-243, c=11682

so using the formula for the vertex coordinate, we get:

x_vertex = - ( - 243)/(2 x 11) = 243 / 22 = 11.045

Since the problem says x measures the number of phones produced in thousands of units, this means that the lowest point of the function, so the lowest cost, happens when 11,045 units are manufactured.

The problem also asks you to find the cost to make this many phones. You have a formula to calculate cost given any value of x, so this is just a matter of plugging in your solution into the function C(x). Remember to use it in units of thousands, though, so 11.045 rather than 11,045.

Answer 2

Minimum occurs when x = -b / 2a, or in this case 243 / 22.

C(243/22) = ?