Question

Morgan has 46 m of fencing to build a four-sided fence around a rectangular plot of land. The area of the land is 130 square meters. Solve for the dimensions (length and width) of the field.

Answer 1

So we have two possible values for the width: W = 16 or W = 7.5.

If W = 16, then L = 130/16 ≈ 8.125. This gives us a perimeter of 2(8.125) + 2(16) = 48.25, which is too much fencing.

If W = 7.5, then L = 130/7.5 ≈ 17.333. This gives us a perimeter of 2(17.333) + 2(7.5) ≈ 46.666, which is close enough to 46. Therefore, the dimensions of the field are approximately L = 17.333 m and W = 7.5 m.

Explanation:

Let the length of the rectangular plot be L, and let the width be W. Then we have:

The perimeter of the plot is 2L + 2W, and this must equal the length of fencing that Morgan has, which is 46 m. So we have the equation:

2L + 2W = 46

The area of the plot is LW, and we know that this equals 130 square meters. So we have the equation:

LW = 130

We can solve for one variable in terms of the other from the second equation, so let's solve for L:

L = 130/W

Now we can substitute this expression for L into the first equation:

2(130/W) + 2W = 46

Simplifying this equation, we get:

260/W + 2W = 46

260 + 2W^2 = 46W

2W^2 - 46W + 260 = 0

Dividing by 2, we get:

W^2 - 23W + 130 = 0

This is a quadratic equation that we can solve using the quadratic formula:

W = [23 ± √(23^2 - 4(1)(130))] / (2(1))

W = [23 ± 9] / 2

So we have two possible values for the width: W = 16 or W = 7.5.

If W = 16, then L = 130/16 ≈ 8.125. This gives us a perimeter of 2(8.125) + 2(16) = 48.25, which is too much fencing.

If W = 7.5, then L = 130/7.5 ≈ 17.333. This gives us a perimeter of 2(17.333) + 2(7.5) ≈ 46.666, which is close enough to 46. Therefore, the dimensions of the field are approximately L = 17.333 m and W = 7.5 m.

Answer 2

Length = 13

Width = 10

Explanation:

Let L = length and W = width of the plot

The area is given by LW and is given as 130 square meters

LW = 130 (1)

The perimeter = 2(L + W) and given as 46 meters(length of fencing)
2(L + W) = 46
L+W = 23 (2)

In equation 1, isolate L by dividing both sides by W
LW/W = 130/W
L = 130/W

Substitute this value for L in equation (2)
L + W = 23
130/W + W = 23

Multiply throughout by W:
130/W x W + W x W = 23x W
130 + W² = 23W

Subtract 130W from both sides:
130 + W² - 23W= 0

Rewrite as

W² - 23W + 130 = 0

This is a quadratic equation which can be solved by factoring

Factoring W² - 23W + 130 = 0
(W - 13)(W-10) = 0

This means W = 13 or W = 10
If W = 13, L = 130/W = 120/13 = 10
If W = 10, L = 130/W= 130/10 = 13

So the possible values for L, W are
L = 13, W = 10

or

L = 10, W =13

Choosing the larger value for L gives us:
L = 13, W = 10