An accessories company finds that the cost, in dollars, of producing $\mathrm{x}$ belts is given by $C(x)=720+39 x-0.065 x^{2}$. Find the rate at which average cost is changing when 172 belts have been produced.
Answer
Final Answer: The rate at which average cost is changing when 172 belts have been produced is approximately \(\boxed{-0.089}\).
Step 1 ：The cost function of producing x belts is given by \(C(x)=720+39x-0.065x^{2}\).
Step 2 ：The average cost function is given by \(A(x) = \frac{C(x)}{x}\).
Step 3 ：The rate at which the average cost is changing is given by the derivative of the average cost function, \(A'(x)\).
Step 4 ：We need to find \(A'(172)\).
Step 5 ：First, we calculate the derivative of the average cost function, \(A'(x) = \frac{39 - 0.13x}{x} - \frac{-0.065x^{2} + 39x + 720}{x^{2}}\).
Step 6 ：Substitute x = 172 into the derivative function, we get \(A'(172) = -0.0893374797187669\).
Step 7 ：Final Answer: The rate at which average cost is changing when 172 belts have been produced is approximately \(\boxed{-0.089}\).
