Question

Select the correct answer from each drop-down menu.A farmer has 144 feet of fending to use to create a rectangular enclosure for his pigs. He decides to use the wall of his barn as one side of the endosure, as shown.The farmer uses funcdon A to model the area of the enclosure, in square feet, when it is x feet wide.
`a(x) = 144x - {2x}^(2)`What width should the farmer use to make the enclosure in order to have the maximum area for his pigs? The best form of the equation for finding the required information is------ the farmer should make the enclosure------- feet wide to have the maximum area for his pigs.

Answer 1

We want to know what value of width will maximize the area of the pigs. This is, we want to find a maximum of the function:

`a(x)=144x-2x^2`

For doing so, we will find its critical points, by the first derivative criteria. Then we find the first derivative:

`a^(\prime)(x)=144-4x`

We equal to zero and solve for x:

`\begin{gathered} 144-4x=0 \\ 144=4x \\ (144)/(4)=x \\ 36=x \end{gathered}`

This means that x=36 is a critical point. Now, we will check if it is a maximum or a minimum using the second derivative test:

`a^(\doubleprime)(x)=-4`

As the second derivative is negative for all x, this means that x=36 is a maximum of the funtion a(x).

On other words, the farmer should use a width of 36 feet to have the maximum area for his pigs.