a) Let's begin by defining the three functions requested. The first two, average cost and average revenue, can be found by dividing the functions we've been given by x.

[Math Processing Error]R¯(x)=50xx=50C¯(x)=x23+750x=x−13+750x

The profit function is defined as the difference between the revenue and cost functions. Likewise, the average profit function is defined as the difference between the average revenue and average cost functions.

[Math Processing Error]P¯(x)=R¯(x)−C¯(x)=50−(x−13+750x)=50−x−13−750x

b) In order to find the rate at which this average cost is changing, we need to differentiate the average cost function. This requires us to use the Power Rule.

[Math Processing Error]C¯′(x)=13x−43−750x2=13x43−750x2

We can now evaluate this for 8 items.

[Math Processing Error]C¯′(8)=13(8)43−750(8)2=1340963−75064=13(16)−11.71875=0.0208333−11.71875≈−11.6979167

This means that the average cost is decreasing by approximately $11.70 per item when 8 items are already produced.