A farmer needs to build a rectangular pen that is 800 square feet and has two bays. What equation can he use to minimize the cost of the fence and what are the dimensions of the pen?
xy = 800
p = 2x+3y = 2x + 2400/x
Now just find x for minimum p.
This assumes two rectangular areas sharing a common fence of length y.
Thanks Steve!
To minimize the cost of the fence, the farmer can use an equation that represents the perimeter of the pen, as the cost of the fence is directly proportional to the length of the fence required.
Let's assume the length of the pen is L and the width is W. Since we know that the pen is rectangular and has an area of 800 square feet, we can write the equation:
L * W = 800
To minimize the cost of the fence, the farmer needs to minimize the perimeter of the pen, which is given by the equation:
P = 2L + 2W
Now, we can substitute the value of L from the first equation into the perimeter equation:
P = 2(800/W) + 2W
To simplify further, we can multiply both terms by W:
P = (1600/W) + 2W
The equation above represents the cost of the fence in terms of W (the width of the pen). To find the width that minimizes the cost, we can take the derivative of this equation with respect to W, set it equal to zero, and solve for W. However, since the equation is non-linear, we may need to use numerical methods or optimization software to find the exact minimum value.
Once the optimal width (W) is found, we can substitute it back into the first equation (L * W = 800) to determine the corresponding length (L) of the pen. These dimensions will give the minimum cost for the fence.