A rancher has 2600 feet of fence to enclose a rectangular region that lies along a straight river.

Question

A rancher has 2600 feet of fence to enclose a rectangular region that lies along a straight river. If no fence is used along the river (see the figure), find the dimensions needed to enclose 5 acres of land. Recall that 1 acre = 43,560 ft².

Answer 1

Let L = Length

Let W = Width

Let P = Perimeter = 2L + 2W

Let A = Area = 5(43560) = 217800 = L(W)

You have 2600 ft to enclose a rectangular region of 5 acres (1 acre = 43560 ft²) that lies along a straight river, with the river serving as one side of the enclosure so

Instead of

P = 2L + 2W

You have

P = L + 2W

2600 = L + 2W

L = 2600 - 2W

Substitute this in the area

217800 = W(2600 - 2W)

217800 = 2600W - 2W²

Move everything over to one side of the equal sign to form a quadratic

217800 - 2600W + 2W² = 0

Divide both sides of the equation above by 2

108900 - 1300W + W² = 0

Factoring

(W - 1210)(W - 90) = 0

W = 90 ft

L = 2600 - 2(90) = 2600 - 180 = 2420 ft; for one long Length and 2 short Widths.When Typically Length represents the longer side.

OR

W = 1210 ft

L = 2600 - 2(1210) = 180 ft; for two large Widths making a for a very wide enclosure with a short length.

I hope this helps.

Maybe you get more value out having the longer length since whatever is in your space has easier and greater access to the water.

Answer 2

There are two ways to set the equation for this problem. They will both lead to the same answer.

I set variable x to be the length of the side parallel to the river. Then, for the other two sides, we have leftover of (2600 - x). One of the two sides will be (2600 - x)/2.

Area of rectangle is the product of length times width.

A = length x width

[5 x 43, 560 ft²] = x (2600 - x)/2.

We simplify and write as quadratic equation:

x² -2600x + 435,600 = 0.

The easiest way to solve this equation is to use quadratic formula with a = 1, b = -2600, c = 435600.

(Steps for solving omitted)

x₁ = 2420

x₂ = 180.

Having two solution means we can construct two different lawns with the same area - one very long and narrow, the other very short and much wider.

If you have questions about solution let me know.

Answer 3

w represents the width

2600 - 2w represents the length

5(43560) = w(2600 - w)

Solution is irrational. Check the data.