# 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?

The dimensions that will maximize the area the total area of the pen will be

Total area with these dimensions:

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

Diagram:

The total length will be

Needed Equations:

Perimeter of this diagram

Total Area

Step 2: Solve for

Step 3: Substitute the equation for

Step 4: Find the derivative of the equation for area.

Step 5: Use the derivative equation in order to find the critical point(s) that maximize the area.

Critical points are when

Since

Step 6: Find the height (

The dimensions that will maximize the area the total area of the pen will be

Step 7: Find the total area of the pen.

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To maximize the total area of the pen, the dimensions of the rectangular pen should be as follows:

Length of the pen (parallel to partitions): ( L = 125 ) feet Width of each section (perpendicular to partitions): ( W = 125 ) feet Height of partitions: ( H = \frac{500}{4} = 125 ) feet

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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