Problem 67 - Algebra and Trigonometry

Question

You have 50 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?

Step-by-Step Solution

Answer

The dimensions that will provide the maximum enclosed area with 50 yards of fencing are length 12.5 yards and width 12.5 yards, this forming a square. The maximum enclosed area possible is 156.25 square yards.

Step 1: Defining the Variables

Let's designate the length of the rectangle as \(x\) and the width as \(y\). Hence, we know that \(2x + 2y = 50\) as this represents the total fencing available for use.

Step 2: Rearranging the Formula

By rearranging the formula above, we can express \(y\) in terms of \(x\). This gives: \(y = 25 - x\).

Step 3: Formulating the Area Function

The area \(A\) of a rectangle is given by the product of length and width. By substituting \(y\) from Step 2 into this formula, an equation for the area in terms of \(x\) is found: \(A = x(25 - x) = 25x - x^2\).

Step 4: Finding the Derivative of the Area Function

Differentiate the formula above to get the maximum and minimum. The derivative \(A' = 25 - 2x\).

Step 5: Solving the Derivative for Zero

To find the 'turning points' of the original function where it may reach a maximum or minimum, set the derivative equal to zero and solve for \(x\): \(25 - 2x = 0\) which leads to \(x = 12.5\).

Step 6: Finding the Corresponding y

Substitute \(x = 12.5\) into the formula from Step 2 to obtain the corresponding \(y = 25 - 12.5 = 12.5\). This shows the rectangle is in fact a square.

Step 7: Verification and Calculation of Maxima

By checking points in the vicinity of \(x = 12.5\), it can be confirmed that this indeed provides a maximum value. Substituting \(x = 12.5\) and \(y = 12.5\) to the area formula yields the maximum area: \(A = 12.5 * 12.5 = 156.25\) square yards.