Math  /  Geometry

QuestionFarmer Ed has 2,000 meters of fencing, and wants to enclose a rectangular plot that borders on a river. If Farmer Ed does not fence the side along the river, what is the largest area that can be enclosed?
The largest area that can be enclosed is \square square meters.

Studdy Solution

STEP 1

What is this asking? We need to find the biggest rectangle Farmer Ed can make with 2,000 meters of fencing, keeping in mind one side is already "fenced" by the river! Watch out! Don't fence all four sides!
Remember, the river acts as one side of the rectangle.

STEP 2

1. Set up the problem
2. Optimize the area

STEP 3

Imagine the river, and Farmer Ed's fence forming three sides of a rectangle.
Let's call the length of the fence parallel to the river xx meters, and the lengths of the two sides perpendicular to the river yy meters each.

STEP 4

Since Farmer Ed has 2,000 meters of fencing, and the total length of the three sides is x+y+yx + y + y, we can write the equation: x+2y=2000x + 2y = 2000.

STEP 5

The area (AA) of the rectangle is given by the product of its sides: A=xyA = x \cdot y.

STEP 6

From our fencing equation (x+2y=2000x + 2y = 2000), we can express xx in terms of yy: x=20002yx = 2000 - 2y.
This helps us work with just one variable!

STEP 7

Now, substitute this expression for xx into our area formula: A=(20002y)y=2000y2y2A = (2000 - 2y) \cdot y = 2000y - 2y^2.
Look! We have a quadratic equation for the area!

STEP 8

The maximum area occurs at the vertex of this parabola.
The yy-coordinate of the vertex is given by y=b2ay = -\frac{b}{2a}, where a=2a = -2 and b=2000b = 2000.
So, y=20002(2)=20004=500y = -\frac{2000}{2 \cdot (-2)} = \frac{2000}{4} = \mathbf{500} meters.

STEP 9

Now we know y=500y = \mathbf{500}, we can plug it back into the equation x=20002yx = 2000 - 2y to find xx: x=20002500=20001000=1000x = 2000 - 2 \cdot 500 = 2000 - 1000 = \mathbf{1000} meters.

STEP 10

Finally, we can calculate the maximum area: A=xy=1000500=500,000A = x \cdot y = 1000 \cdot 500 = \mathbf{500,000} square meters.

STEP 11

The largest area that can be enclosed is **500,000** square meters.

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