Word problem involving the maximum or minimum of a quadratic function

Question

A vehicle factory manufactures cars. The unit cost C (the cost in dollars to make each car) depends on the number of cars made. If x cars are made, then the unit cost is given by the function

 How many cars must be made to minimize the unit cost? Do not round your answer.

Answer 1

Question: A vehicle factory manufactures cars. The unit cost C (the cost in dollars to make each car) depends on the number of cars made. If x cars are made, then the unit cost is given by the function

How many cars must be made to minimize the unit cost? Do not round your answer.

Answer: Notice that the function C(x) is quadratic because it is a second-degree polynomial. See the x² term? That's how we know. Also, see how the number attached to the x² term, the 0.9, is positive? This is a positive leading coefficient, so the graph of this function (which is a parabola, a U-shape), is right-side up, a typical "U." Notice how the bottom of the parabola looks like the bottom, the floor of a valley. It's a low point, a minimum. Also, note that we're plugging x-values into C(x) to get a parabola. We're plugging in a number of cars made and getting back a unit cost. At the lowest point of this parabola, at its vertex, we're going to get back the lowest, the minimum unit cost.

So all we need is a formula to get the vertex of this parabola!

First, what's standard form for a quadratic equation? ax² + bx + c = 0.

Find what a, b, and c are. These should be a = 0.9, b = -378, and c = 47479.

You might have seen this formula before: x = - (b/2a). This is the formula for the x-coordinate of the vertex of a parabola, and it's also the formula for the axis of symmetry of a parabola. Plug in values to get back x (here, the number of cars made):

x = - (b/2a)

x = - ((-378)/(2*(0.9)))

x = - (-378/1.8)

x = - (-210)

x = 210

210 cars need to be made to minimize the unit cost. If you're curious as to what this minimum unit cost actually *is*, plug 210 back into the original equation wherever you see x and simplify:

C(x) = 0.9x² - 378x + 47479

C(210) = 0.9(210)² - 378(210) + 47479

C(210) = 0.9(44100) - 79380 + 47479

C(210) = 39690 - 79380 + 47479

C(210) = 7789

So the vertex of this parabola is (210, 7789).

The unit cost is minimized at 210 cars made, where the unit cost (the cost to make each car) is $7,789.

Answer 2

Could use calculus. Take 1st derivative and set = 0 (slope of parabola = 0 at the vertex, which is the minimum in this case).

dy/dx = 0 = 1.8x - 378 x = 210