A farmer has 120 feet of fencing available to build a rectangular pen for her pygmy goats. She wants to give them as much room as possible to run. What are the dimensions of the rectangular pen with the largest area?

A rectangular room has a perimeter of 36 feet.One side is 10 feet long.How long is side s?

To find the dimensions of the rectangular pen with the largest area, we need to consider the relationship between the length and the width of the pen. Let's assume that the length of the pen is "L" feet and the width is "W" feet.

Since the perimeter of a rectangle is the sum of all its sides, we know that:

2L + 2W = 120

Simplifying this equation, we get:

L + W = 60

Now, we want to maximize the area of the rectangular pen, which is given by the formula:

Area = Length * Width

Substituting "60 - W" for "L" in the area formula, we get:

Area = (60 - W) * W

To find the maximum area, we can take the derivative of the area function and set it equal to zero. However, since this is a simple quadratic equation, we can use a different approach.

To maximize a quadratic equation (Area = (60 - W) * W), we need to find the vertex of the parabola. The vertex of a parabola in the form ax^2 + bx + c is given by the formula:

x = -b/2a

In our case, the equation is W(60 - W). So, a = -1 and b = 60.

x = -b/2a = -60/(2 * -1) = 30

Therefore, the width of the rectangular pen that results in the largest area is 30 feet.

Substituting this value back into the equation L + W = 60, we find:

L + 30 = 60

L = 60 - 30 = 30

Hence, the dimensions of the rectangular pen with the largest area are 30 feet by 30 feet.

30 * 30 = 900 sq. ft.

A rectangular room has a perimeter of 36 feet.One side is 10 feet long.How long is side s?