Question

{35 POINTS} A factory produces t-shirts. The production cost, C(x), for x t-shirts can be modeled by a quadratic function.

Each of the following functions is a different form of the quadratic model for the situation given above. Which form would be the most helpful if attempting to determine the number of t-shirts that would minimize cost?
A.
C(x) = 0.02(x2 - 720x + 144,600)

B.
C(x) = 0.02(x - 360)2 + 300

C.
C(x) = 0.02x2 - 14.4x + 2,892

D.
C(x) = 0.02x(x - 720) + 2,892

Answer 1

A

Explanation:

Please use " ^ " to denote exponentiation: x^2 (not x2).

To determine the number of t-shirts that would minimize cost, we'd want to have a quadratic function graph that opens up, and to determine the vertex (which point would also be the minimum cost).

Equation A is the one we want. Why? because we can ignore the coefficient 0.02 in determining the minimum cost. We see immediately that the coefficient of the x^2 term is a = 1 and that of the x term is b = -720.

Recall that the axis of symmetry passes through the vertex / minimum, and has equation

720

x = -b / (2a). With a = 1 and b = -720, x = - --------- = 360

2

Again, A is the correct answer. The miminim cost occurs at x = 360 (shirts).

Answer 2

B. C(x) = 0.02(x - 360)2 + 300

Explanation:

The minimum of a quadratic function is found at its vertex. The vertex form of the equation lets you read the coordinates of the vertex directly, so that is the form most helpful for finding minimum cost.

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Perhaps the next most helpful form is that of option D, sometimes called "intercept form." The vertex is located halfway between the intercepts, so will be halfway between 0 and 720. Of course, the vertex location at 360 is read directly from the vertex form of option B.