Yes, I can help with this problem!

We are given that the perimeter of the rectangle is $P = 90$ ft, and we need to find:

a. The largest possible area using the given formula:
$A = \left( \frac{P}{4} \right)^2$

Let's solve this step by step.

Part a: Maximum Area

Using the formula for the maximum area of a rectangle when the perimeter is fixed, we calculate the area $A$ as:

$A = \left( \frac{P}{4} \right)^2$

Substituting $P = 90$ ft:

$A = \left( \frac{90}{4} \right)^2 = \left( 22.5 \right)^2 = 506.25 \, \text{square feet}$

So, the largest possible area is 506.25 square feet.

Part b: Maximum Dimensions

The rectangle that gives the maximum area for a fixed perimeter is actually a square. Since the perimeter is 90 feet, and for a square, all sides are equal, we can find the side length by dividing the perimeter by 4:

$\text{Side length} = \frac{P}{4} = \frac{90}{4} = 22.5 \, \text{ft}$

Thus, the maximum dimensions are 22.5 ft by 22.5 ft, forming a square.

Would you like further details or explanations on any part?

5 Related Questions:

1. How would the area change if the perimeter were increased to 100 feet?
2. What would happen to the area if the rectangle were not a square but a longer, narrower shape?
3. Can you derive the formula $A = \left( \frac{P}{4} \right)^2$ from first principles?
4. How does the shape of the rectangle affect its area when the perimeter is constant?
5. How can you apply calculus to confirm that a square gives the maximum area for a given perimeter?

Tip:

For any fixed perimeter, the shape that maximizes the area is always a square. This is a useful principle in geometry and optimization problems.