The total cost (in hundreds of dollars) to produce $x$ units of perfume is $C(x)=\frac{5 x-9}{9 x+4}$. (a) Find the average cost function. (b) Find the marginal average cost function. (c) Find the average cost and the marginal average cost for a production level of 40 units. Interpret your results. (a) The average cost function is $\overline{\mathrm{C}}(\mathrm{x})=\square$. (b) The marginal average cost function is $\overline{\mathrm{C}}^{\prime}(\mathrm{x})=\square$. (c) The average cost for 40 units is $\$ \square$ per unit. The marginal average cost for 40 units is $\$ \square$. (Round to the nearest cent as needed.) Interpret your results. When 40 units are produced, the average cost is $\$ \square$ per unit and the average cost is $\square$ at a rate of about $\$ \square$ per unit. (Round to the nearest cent as needed.)

Step 1 ：The average cost function is \(\overline{C}(x) = \frac{C(x)}{x}\). Substituting the given cost function, we get \(\overline{C}(x) = \frac{\frac{5x-9}{9x+4}}{x} = \frac{5x-9}{x(9x+4)} = \frac{5 - \frac{9}{x}}{9 + \frac{4}{x}}\).

Step 2 ：The marginal average cost function is the derivative of the average cost function. To find this, we first need to find the derivative of the numerator and the denominator of the average cost function separately.

Step 3 ：The derivative of the numerator, \(5 - \frac{9}{x}\), is \(0 - (-\frac{9}{x^2}) = \frac{9}{x^2}\).

Step 4 ：The derivative of the denominator, \(9 + \frac{4}{x}\), is \(0 - (-\frac{4}{x^2}) = \frac{4}{x^2}\).

Step 5 ：Using the quotient rule for differentiation, which states that the derivative of \(\frac{u}{v} = \frac{vu' - uv'}{v^2}\), where \(u\) is the numerator, \(v\) is the denominator, \(u'\) is the derivative of the numerator, and \(v'\) is the derivative of the denominator, we get:

Step 6 ：\(\overline{C}'(x) = \frac{(9 + \frac{4}{x})(\frac{9}{x^2}) - (5 - \frac{9}{x})(\frac{4}{x^2})}{(9 + \frac{4}{x})^2} = \frac{81 + 36 - 20 + \frac{36}{x}}{(9x + 4)^2}\).

Step 7 ：To find the average cost and the marginal average cost for a production level of 40 units, we substitute \(x = 40\) into the average cost function and the marginal average cost function.

Step 8 ：The average cost for 40 units is \(\overline{C}(40) = \frac{5 - \frac{9}{40}}{9 + \frac{4}{40}} = \frac{4.775}{9.1} = 0.525\) hundred dollars per unit, or $52.50 per unit.

Step 9 ：The marginal average cost for 40 units is \(\overline{C}'(40) = \frac{81 + 36 - 20 + \frac{36}{40}}{(9*40 + 4)^2} = \frac{97 + 0.9}{1936} = 0.0505\) hundred dollars per unit, or $5.05 per unit.

Step 10 ：\(\boxed{\text{Interpretation: When 40 units are produced, the average cost is $52.50 per unit and the average cost is increasing at a rate of about $5.05 per unit.}}\)