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}\\)
(a) use a graphing utility to determine the number of riding lawn mowers to produce in order to minimize average cost.
(b) what is the minimum average cost?

(a) the average cost is minimized when approximately \\(\square\\) lawn mowers are produced per hour.
(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 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 method, $x \approx 9$

Step4: Calculate min average cost

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

Answer:

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