How many items should be produced to minimize the cost? Please help

Question

Please help........Suppose that the cost function for the production of a particular item is given by the function C(x)=2x2−320x+12,020, where x represents the number of items. How many items should be produced to minimize the cost? Explain your answer.

Answer 1

divideCost by 2

(1/2)C=x²-160x+6,010

Take the 1st derivative and set equal to zero to find local max or min

C'=2x-160=0 2x=160 x=80

At x=80 (1/2)C=6400-12,800+6,010=-390 C=-780 or rather profit =780 or cost = negative 780

The cost function is a parabola, a quadratic equation. The slope of the parabola is zero

at its minimum point. the 1st derivative is the slope of the tangent to the parabola.

If you produce more or less than 80, the cost rises as the parabola rises.

The point (x,C) = (80,-780) is the vertex of the parabola

But you may not know calculus. So try algebra, using either the quadratic equation or completing the square with the cost function set equal to zero. You get 2 answers. Take the average of their x values, that will be the x value of the minimum cost.

x²-160x+6010=0 Take half of 160=80 Square it to get 6,400, then reduce this equation

to( x²-160x+6,400) + (6010-6400). In other words, add and subtract 6,400 to get

(x-80)² -390 = 0 Then put the constant term on the right side and take square roots of both sides: x-80= plus or minus (390)^1/2 x=80 plus the square root of 390 is an x-intercept, a break even point where costs = 0. x=80 minus the square root of 390 is the other x-intercept where costs equal zero. The midpoint between those two intercepts is the vertex of the parabola or minimum cost, x=80

20 squared is 400, so the square root of 390 is slightly less than 20, maybe about 19. The break even points, where costs equal revenue or profits = 0 is about 80-19 and 80+19 or 61 and 99 units. The midpoint, is 80, with minimum cost or maximum profit. the parabola is symmetric about the line x=80.

If you ever get the same solution by doing the problem 2 alternative ways, you feel very confident about the answer. Calculus and algebra lead to the same solution, produce 80 items to minimize cost.

You can also just graph the equation carefully and see where the minimum point on the parabola is.