Question

A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 800 meters of wire at your disposal, what is the largest area you can enclose, and what are its dimensions?

Answer 1

Answer: The largest area that can be enclosed is 80,000 square meters (80000 m²) and the dimensions are 400 meters by 200 meters.

Step-by-step explanation: The total length of electric available is 800 meters, so we already know that the perimeter is 800 meters in total. Also we have been told that the plot of farmland would be bounded on one side by a river, which implies that the perimeter fencing will not include one of the four sides of the rectangular shape.

Rather than use the conventional formular of calculating a perimeter (Per = 2{L + W}), we shall apportion the perimeter fencing into three sides. Knowing that it is a rectangular shape, we can determine that one side is longer than the other (that is not all four sides have equal measurement).

The sides can be determined as;

800 = 2 : 1 : 1

So one side shall be

x/800 = 2/4

x/800 = 1/2

2x = 800

Divide both sides of the equation by 2

x = 400

The other two sides which measure the same length can be derived as;

y/800 = 1/4

4y = 800

y = 200

Therefore one side of the fencing would measure 400 meters and the other two sides would measure 200 meters each. And the total area that would be enclosed is given as

Area = L x W

Area = 400 x 200

Area = 80000 square meters

Answer 2

Length = 400 m

Width = 200 m

Area = 80,000 m2

Explanation:

Let's call the length L and the width W. Then, we have that the perimeter will be:

L + 2W = 800.

We want to maximize the area, that is:

A = L*W

so, from the first equation, we have:

L = 800 - 2W

Using this value in the area equation, we have:

A = (800 - 2W)*W = 800W - 2W^2

To find the maximum, we just need to find the vertix of the quadratic equation, and then apply the value to find the area:

W_vertix = -b/2a

Where a and b are coefficients of the quadratic equation (in our case, a = -2 and b = 800)

W_vertix = -800/(-4) = 200 m

A = (800 - 400)*200 = 80000 m2

L = 800 - 2W = 800 - 400 = 400 m

So the dimensions and area are:

Length = 400 m

Width = 200 m

Area = 80,000 m2