Business average cost.

Question

Daniels digital documents finds that the cost, in dollars, of producing x boxes of trifold brochures is given by: C(x)=400+6√x.

 

find the rate at which the average cost is changing when 25 boxes of trifold brochures have been produced.

Answer 1

To answer this question, I will assume that this situation can be modeled as continuous (as opposed to discrete), which allows us to use calculus.

 

First, if we want to find the rate of change of the average cost at some value of x, we need to obtain an expression for the average cost in terms of x. To obtain such an expression, we will need to integrate C(x) from 0 to x and then divide the answer by the size of the interval over which we integrated (x - 0).  This yields

 

(400x + 6(2/3)x^3/2)/x = 400 + 4x^1/2

 

This expressions tells you the average value of C(x) over the interval 0 to x.  Now, we differentiate this expression to find the rate of change of the average cost, then plug in 25 to get the answer.

 

The derivative is 2x^-1/2, so after we plug in 25 for x, we get

 

2(25)^-1/2 = 2/5

 

So, the rate at which the average cost is changing is 2/5.