Chapter 2.3, Problem 75AYU

a.

To determine

To graph: The given function using a graphing utility.

Explanation of Solution

Given Information: Given function is $\overline{C}(x)=0.3x^2+21x-251+\frac{2500}{x}$ , which represents which represents average cost per hour (in dollars) of producing $x$ riding lawn mowers.

Graph:

  

b.

To determine

To find: The number of riding lawn mowers to produce in order to minimize average cost.

Answer to Problem 75AYU

  $9.659$ riding lawn mowers are needed to minimize average cost.

Explanation of Solution

Given Information: Given function is $\overline{C}(x)=0.3x^2+21x-251+\frac{2500}{x}$ , which represents which represents average cost per hour (in dollars) of producing $x$ riding lawn mowers.

Calculation:

To minimize the average cost,

  $\begin{array}{l}\overline{C}\text{'}(x)=0\\ \Rightarrow 0.6x+21-\frac{2500}{x^2}=0\\ \Rightarrow 0.6x^3+21x^2-2500=0\\ \Rightarrow x=-30.529598,-14.129557,9.659155\end{array}$

Since number of lawn mowers cannot be negative, $\therefore x=9.659155$

Thus, $9.659$ riding lawn mowers are needed to minimize average cost.

c.

To determine

To find: The minimum average cost.

Answer to Problem 75AYU

The minimum average cost is $238.654dollars$ .

Explanation of Solution

Given Information: Given function is $\overline{C}(x)=0.3x^2+21x-251+\frac{2500}{x}$ , which represents which represents average cost per hour (in dollars) of producing $x$ riding lawn mowers.

Calculation:

Putting $x=9.659155$ in $\overline{C}(x)=0.3x^2+21x-251+\frac{2500}{x}$ ,

  $\begin{array}{l}{\overline{C}}_{\min }=0.3{(}^{9.659}+21(9.659)-251+\frac{2500}{9.659}\\ =238.654\end{array}$

Thus, the minimum average cost is $238.654dollars$ .