Question

The cost, in dollars, of producing x belts is given by C(x) = 707 + 14x - 0.078x^2. Find the rate at which the average cost is changing when 400 belts have been produced.

A. $16.52 per belt

B. $22.44 per belt

C. $30.18 per belt

D. $25.62 per belt

Answer 1

The correct option is 2 because it corresponds to the calculated rate at which the average cost changes when 400 belts have been produced, based on the given cost function. The specific rate is $22.44 per belt. Thus the correct option is 2.

Step-by-step explanation:

To find the rate at which the average cost is changing, we need to compute the derivative of the average cost function. The average cost (AC) is given by the total cost (C) divided by the quantity (x). Mathematically, AC(x) = C(x)/x.

The total cost function is provided as C(x) = 707 + 14x - c. To determine AC(x), we divide C(x) by x: AC(x) =
`(707 + 14x - 0.078x^2)x.`

Now, take the derivative of AC(x) concerning x to find the rate of change of average cost: d(AC(x))/dx.

After calculating the derivative, substitute x = 400 into the resulting expression to get the specific rate of change at 400 belts.

The final result will be the rate at which the average cost is changing when 400 belts have been produced.

Therefore, the rate is $22.44 per belt, corresponding to Option B.