Question

Marcus has 72 feet of fencing. He wants to build a pen with the largest possible area. What should the dimensions of the rectangle be?

Answer 1

It should be a square with 18 feet on each side.

To find the dimensions of the rectangle that would result in the largest possible area for the pen, we can use optimization techniques. The formula for the area of a rectangle is given by A = length × width.

Let's assume the length of the rectangle is L feet, and the width is W feet.

We are given that Marcus has 72 feet of fencing available. The perimeter of the rectangle would be the sum of all four sides, which is twice the length plus twice the width:

Perimeter = 2L + 2W

Since Marcus has 72 feet of fencing, we can write the equation:

2L + 2W = 72

Now, we want to find the maximum possible area of the rectangle. The area (A) of the rectangle is given by:

A = L × W

To maximize the area, we can solve for one of the variables in terms of the other using the perimeter equation and then substitute it into the area equation.

Let's solve for L in terms of W using the perimeter equation:

2L + 2W = 72

2L = 72 - 2W

L = 36 - W

Now, we can substitute L = 36 - W into the area equation:

A = L × W

A = (36 - W) × W

A = 36W - W^2

The area of the rectangle, A, is a quadratic function of W, which represents the width. To find the maximum area, we need to find the value of W that maximizes the quadratic function. Since the coefficient of W^2 is negative, the graph of the function will be an inverted parabola, and the maximum area will occur at the vertex of the parabola.

The x-coordinate of the vertex of a parabola given by the equation f(x) = ax^2 + bx + c is given by x = -b / (2a).

In our case, a = -1 (coefficient of W^2), and b = 36 (coefficient of W). So the x-coordinate of the vertex is:

W = -b / (2a) = -36 / (2 × -1) = 36 / 2 = 18

Now that we have the value of W, we can find the corresponding value of L using the perimeter equation:

2L + 2W = 72

2L + 2(18) = 72

2L + 36 = 72

2L = 72 - 36

L = 36 / 2

L = 18

So, the dimensions of the rectangle that result in the largest possible area are:

Length (L) = 18 feet

Width (W) = 18 feet

Therefore, Marcus should build a square pen with each side measuring 18 feet to have the largest possible area with 72 feet of fencing.

I hope this helped!

~~~Harsha~~~

Answer 2

18 ft by 18 ft

Explanation:

Let one side of the rectangular pen measure x.

Let the other side of the rectangular pen measure y.

We need to express the length and width in terms of x.

P = 2x + 2y

2x + 2y = 72

x + y = 36

y = 36 - x

Length = x

Width = 36 - x

Now we find an expression for the area of the rectangle.

A = xy

A = x(36 - x)

A = -x² + 36x

y = -x² + 36x is an inverted parabola, so it has a maximum value.

The parabola is inverted (it opens downward). It has symmetry about the vertical axis. The maximum value occurs at the average of the x-intercepts of the equation of the parabola.

-x² + 36x = 0

x(x - 36) = 0

x = 0 or x = 36

The maximum occurs at x = (0 + 36)/2 = 18

P = 2x + 2y

2(18) + 2y = 72

2y = 36

y = 18

x = y = 18

The maximum area occurs when both the length and width are 18 ft, so the rectangle is a square with 18-ft sides.