A farmer will enclose a rectangular region along a river. He will not fence along the river.
1. If he uses 800ft of fencing, express the length of the rectangle in terms of w, the width.
2. Express the area of the enclosed rectangle in terms of the width, w.
3. What should the width be so that the enclosed area is 80,000 sqft
L = (800-2w)/2 = 400-w
a = w(400-w)
w(400-w) = 80000
solve for w as usual
oops. I missed the part about the river.
So, assuming the length is along the river,
L = 800-2w
If the width is along the river,
L = (800-w)/2
and proceed as above
To answer these questions, we need to understand the relationship between the dimensions of the rectangle and the amount of fencing used.
1. Let's label the length of the rectangle as L and the width as W. Since the farmer will not fence along the river, the total length of fencing required will be the sum of the lengths of the remaining three sides:
Length of fencing = Length of the rectangle + 2 * Width of the rectangle.
Given that the farmer uses 800 ft of fencing, we can write the equation as:
800 = L + 2W.
Now, if we want to express the length of the rectangle (L) in terms of the width (W), we can rearrange the equation:
L = 800 - 2W.
Therefore, the length of the rectangle is expressed as L = 800 - 2W.
2. The formula to calculate the area of a rectangle is given by:
Area = Length * Width.
So, to express the area of the enclosed rectangle in terms of the width (W), we substitute the expression for the length of the rectangle into the formula for the area:
Area = (800 - 2W) * W.
Therefore, the area of the rectangle is expressed as Area = (800 - 2W) * W.
3. We want to find the width (W) that will result in an enclosed area of 80,000 sqft. Since we have the expression for the area in terms of the width, we can set it equal to 80,000 and solve for W:
(800 - 2W) * W = 80,000.
Distribute the W:
800W - 2W^2 = 80,000.
Rearrange in standard form:
2W^2 - 800W + 80,000 = 0.
This quadratic equation can be solved using factoring, completing the square, or using the quadratic formula to find the value(s) of W that satisfy the equation. In this case, let's solve it by factoring:
2(W^2 - 400W + 40,000) = 0.
Divide both sides by 2:
W^2 - 400W + 40,000 = 0.
Factor the quadratic equation:
(W - 200)(W - 200) = 0.
From this, we find that the only solution for W is 200 ft.
Therefore, the width should be 200 ft in order for the enclosed area to be 80,000 sqft.