Please help ASAP! Algebra word problem....

Question

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

Given C(x)=2x²−320x+12,020, where x represents the number of items, to find out how many items should be produced to minimize the cost, we can solve for the vertex. This is a quadratic equation written in the standard form ax²+bx+c and we know in this equation, a=2, b=-320, and c=12,020. Since a>0, the parabola opens upward which means it has a minimum at its vertex (the parabola looks like U so it would have a minimum point which is called the vertex).

If a quadratic function is in standard form, we can find the x-coordinate of the vertex by using the vertex formula x=-b/2a so lets apply it:

x = -(-320) / 2(2) = 320 / 4 = 80

We can also find the y-coordinate of the vertex by plugging this x value back into the equation. However, we don't need to for this question because it asks us how many items should be produced to minimize the cost and items is x which we just found. If it asked us for the cost, then we would need to find the y by plugging in x into the equation.

To conclude, the cost is minimized at x = 80; meaning producing 80 items results in the minimum cost.

Answer 2

Divide by 2

C=x²-160x+6,010

Take the first derivative

C'=2x-160 set it equal to zero 2x-160=0

divide by 2: x-80=0 or x=80

The cost function is an equation which graphed is a parabola. It's minimum point is where the slope equals zero. That's the point where the first derivative equals zero.

At x=80 cost = 6400-12800+6,010= -390 This negative cost is profit.

Or you could algebraically solve the quadratic equation a=2,

b=-320

c=12020

plug into the quadratic formula x=1/2a(-b + or - square root of b²-4ac)

=1/4(320 + or - square root of 320² - 4(2)(12020)) = 1/4(320 + or - square root of 6,240)

=80 + or - square root of 6240

=80 + or - almost 79 = approximately 1 or 159

80 is the midpoint, with 1 or 159 as the break even points where cost=revenues and profit =0

80 is maximum profit or minimum cost

Plug 80 into the original cost equation, and you get the minimum cost or maximum profit.