Question

the average cost per hour in dollars of producing x riding lawn mowers is given by the following.\\(\bar{c}(x)=0.2x^2 + 20x - 242 + \frac{2200}{x}\\)\\(\text{(a) use a graphing utility to determine the number of riding lawn mowers to produce in order to minimize average cost}\\)\\(\text{(b) what is the minimum average cost?}\\)\\(\text{(a) the average cost is minimized when approximately } \square \text{ lawn mowers are produced per hour.}\\)\\(\text{(round to the nearest whole number as needed.)}\\)

Explanation:

Step1: Find derivative of $\bar{C}(x)$

$\bar{C}'(x) = 0.4x + 20 - \frac{2200}{x^2}$

Step2: Set derivative to 0

$0.4x + 20 - \frac{2200}{x^2} = 0$
Multiply through by $x^2$: $0.4x^3 + 20x^2 - 2200 = 0$
Divide by 0.4: $x^3 + 50x^2 - 5500 = 0$

Step3: Solve for positive x

Using numerical/graphical methods, the positive real root is approximately $x \approx 9$

Step4: Calculate min average cost

Substitute $x=9$ into $\bar{C}(x)$:
$\bar{C}(9) = 0.2(9)^2 + 20(9) - 242 + \frac{2200}{9}$
$\bar{C}(9) = 0.2(81) + 180 - 242 + \frac{2200}{9}$
$\bar{C}(9) = 16.2 + 180 - 242 + 244.44$
$\bar{C}(9) \approx 198.64$

Answer:

(a) 9
(b) $\$198.64$