Question

Build a rectangular pen with three parallel partitions (meaning 4 total sections) using 500 feet of fencing. What dimensions will maximize the total area of the pen?

Answer 1

The dimensions that will maximize the area the total area of the pen will be `125ft.` `by` `50ft.`
Total area with these dimensions: `6,250ft.^2`

Explanation:

Step 1: Set up a picture/diagram to help answer the question and write out the needed equations.

Diagram:

The total length will be `x` and the height will be `y`.
Needed Equations:
Perimeter of this diagram `=500=2x+5y`
Total Area`=A=xy`

Step 2: Solve for `y` using the equation `500=2x+5y`
`5y=500-2x`
`y=100-2/5x`

Step 3: Substitute the equation for `y` into the function for area.
`A=x(100-2/5x)`
`A=-2/5x^2+100x`

Step 4: Find the derivative of the equation for area.
`A'=-4/5x+100`

Step 5: Use the derivative equation in order to find the critical point(s) that maximize the area.
Critical points are when `A'=0` and when `A'` does not exist. It is also good to check the endpoints of an equation in order to check for a maximum or minimum.
Since `A'` always exists, only find where `A'=0` (there will be no endpoints to check since this is a pen).

`0=-4/5x+100`
`-100=-4/5x`
`100=4/5x`
`x=125`

`A'` is positive when `x<125` and `A'` is negative when `x>125`, therefore meaning that x=125 is a maximum. Since this value is a maximum, the area is maximized when the total length is 125ft.

Step 6: Find the height (`y`) when `x=125`.
`500=2(125)+5y`
`5y=250`
`y=50`
The dimensions that will maximize the area the total area of the pen will be `125ft.` `by` `50ft.`

Step 7: Find the total area of the pen.
`A=xy`
`A=(125)(50)`
`A=6,250ft.^2`