Question

A farmer w/ 750ft of fencing wants to enclose a rectangular area then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens?

Answer 1

`A = 14,062.5ft^2`

Explanation:

STEP 1: First, we should write down what we know. We know `A = xy` and the perimeter equals 750ft. Using a diagram, we can form the equation `750 =5x + 2y`

Step 2: Solve for y in terms of x. You should get `y=(-5/2)x +375`. This information is going to help us get our Area equation in terms of one variable only (just in terms of x).

Step 3: Using the relationship you found between x & y in Step 2, put your Area equation in terms of x only. You should get `A = xy = x*[(-5/2)x +375] = (-5/2)x^2 + 375x`

Step 4: Derive your area equation, and you should get A' = `-5x +375`

Step 5: Find the critical numbers to get your potential minimums & maximums. Set A' = 0 (since A' exists everywhere, we do not need to worry about A' DNE) and solve for x. You should get x = 75

Step 6: Find y but plugging 75 in for x in your perimeter equation. You should find y = 187.5

Step 7: Plug in your x and y in the Area equation, and you should arrive at your answer of 14,062.5 `"ft"^2`