Austin is building a building a rabbit pen with 25 feet of fence. What are the dimensions of the rectangle he should build to have the greatest possible area?

I need an answer to question

I generalized that a square would have the greatest area. Length and width would be 5, so since a square has equal sides, 4x5= 20 ft, the perimeter and 4x5= 20 sq ft, the greatest area.

me to

Well, Austin must be hopping with excitement to build a rabbit pen! To maximize the area, he should build a rectangle with equal sides, like a square. Since he has 25 feet of fence, we can divide it equally among the four sides of the square. That means each side of the square would be 25/4 feet long. So, the dimensions of the rectangle he should build are approximately 6.25 feet by 6.25 feet. Just make sure those bunnies don't multiply like rabbits!

To find the dimensions that will result in the greatest possible area for the rabbit pen, we can use the concept of optimization.

Let's assume the rabbit pen has a rectangle shape, with width (W) and length (L). Since it's a rectangle, there are two sides with a length equal to W and two sides with a length equal to L.

According to the problem, Austin has 25 feet of fence to use. The total amount of fence used would be the sum of all four sides of the rectangle, which is given as 25 feet.

Mathematically, we can express this as an equation:

2W + 2L = 25

Now, we need to express the area of the rectangle (A) in terms of W and L. The area of a rectangle is calculated by multiplying its length and width:

A = L * W

To determine the dimensions that maximize the area, we need to optimize the equation. We can solve for one variable in terms of the other using the constraint equation mentioned earlier, and then substitute it into the area equation.

From the constraint equation:
2W + 2L = 25

We can divide both sides by 2 to simplify:
W + L = 12.5

Solving for L in terms of W:
L = 12.5 - W

Now we substitute L into the area equation:
A = L * W
A = (12.5 - W) * W

Now we have the area equation in terms of a single variable, W. We can graph this equation or use calculus to find the maximum value of A, which corresponds to the dimensions with the greatest possible area.

Let's differentiate the area equation with respect to W to find the critical points:

dA/dW = 12.5 - 2W

Setting dA/dW = 0 to find the critical points:
12.5 - 2W = 0
2W = 12.5
W = 6.25

Substituting W = 6.25 back into the constraint equation to find L:
L = 12.5 - W
L = 12.5 - 6.25
L = 6.25

Therefore, the dimensions that result in the greatest possible area for the rabbit pen are a width of 6.25 feet and a length of 6.25 feet.

max area will be square, or length=width.

so....Length=Width= 25/4 feet