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Can you answer this using this formula A=(P/4)^2

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.

Suppose the perimeter of a rectangle is 90 ft. What is the largest possible area you may have based on this measurement? What are the maximum dimensions of the rectangular field?

Discover how to find the maximum possible area of a rectangle with a given perimeter of 90 ft. Learn why the rectangle that maximizes the area is actually a square, and use the formula A = (P/4)^2 to calculate the largest area. Suitable for students in Grades 8-10.

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