Question

A farmer is building a rectangular pen along the side of a barn for animals. The barn will serve as one side of the pen. The farmer has 120 feet of fence to enclose an area of 1512 square feet and wants each side of the pen to be at least 20 feet long.Find the dimensions of the pen. How would I put this into my graphing calculator to solve?

Answer 1

The dimension of pen is 42 feet by 36 feet

Solution:

Let "x" be the width of pen

Let "y" be the length

Farmer wants each side of the pen to be at least 20 feet long

$x\geq 20$

The farmer has 120 feet of fence to enclose an area of 1512 square feet

The amount of fencing is equal to the perimeter of fence which is 2 times the width plus only one length since the other side (length) is along the barn

2x + y = 120

Therefore,

y = 120 - 2x

The area of barn is given as 1512 square feet

The area of rectangle is given as:

$Area = length \times width$

$1512 = x \times y\\\\1512 = x \times (120-2x)\\\\1512 = 120x - 2x^2\\\\2x^2-120x + 1512 = 0$

Divide the entire equation by 2

$x^2-60x + 756 = 0$

Factor the left side of equation

$x^2-60x+756 = 0\\\\(x-18)(x-42) = 0$

Therefore, we get two values of "x"

x = 18 or x = 42

Since, $x\geq 20$

Therefore, x = 42 is the solution

Thus, width = x = 42 feet

Length = y = 120 - 2x

y = 120 - 2(42)

y = 120 - 84

y = 36

Thus the dimension of pen is 42 feet by 36 feet

Answer 2

The area of a shape is the amount of space it occupies.

The dimension of the pen is 42 by 36 feet.

The perimeter is given as:

$\mathbf{P = 120}$

Because one of the sides does not need fencing, the perimeter would be:

$\mathbf{P = 2x + y}$

Make y the subject

$\mathbf{y = P - 2x}$

Substitute 160 for P

$\mathbf{y = 120 - 2x}$

The area of a pen is:

$\mathbf{A = xy}$

Substitute $\mathbf{y = 120 - 2x}$

$\mathbf{A = x(120 -2x)}$

Substitute 1512 for Area

$\mathbf{x(120 -2x) = 1512}$

Open brackets

$\mathbf{120x -2x^2 = 1512}$

Rewrite as:

$\mathbf{2x^2 -120x + 1512 = 0}$

Divide through by 2

$\mathbf{x^2 -60x + 756 = 0}$

Expand

$\mathbf{x^2 -18x - 42x + 756 = 0}$

Factorize

$\mathbf{(x -18)(x - 42) = 0}$

Solve for x

$\mathbf{x =18 \ or\ x = 42}$

The dimension must be at least 20.

So, we have:

$\mathbf{x = 42}$

Recall that:

$\mathbf{y = 120 - 2x}$

This gives:

$\mathbf{y = 120 - 2 \times 42}$

$\mathbf{y = 36}$

Hence, the dimension of the pen is 42 by 36 feet.

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