SOlve you have 78 feet of fencing to enclose a rectangular pasture by a barn. If you do not fence the side along the barn, find the length and width of the lot that will maximize the area
78/3 = 26 feet along each side of the square.
To solve this problem, we'll use mathematical optimization. The goal is to find the length and width of the lot that will maximize the area.
Let's start by defining our variables. Let:
- L be the length of the rectangular lot
- W be the width of the rectangular lot
We can assume that the side of the rectangular lot along the barn does not require any fencing, so we only need to consider three sides for fencing.
Since we have a total of 78 feet of fencing available, we can write the equation for the perimeter of the rectangular lot as:
2L + W = 78
We need to rewrite this equation to express one variable in terms of the other. Let's solve it for L:
2L = 78 - W
L = (78 - W)/2
Now, we can use the area formula for a rectangle:
Area = Length × Width
Substitute the expression for L into the area formula:
Area = [(78 - W)/2] × W
Area = (78W - W^2)/2
We now have the area in terms of the width of the lot. To maximize the area, we need to find the critical points of the function.
To find the critical points, we take the derivative of the area function with respect to W and set it equal to zero:
d/dW [(78W - W^2)/2] = 0
Differentiating the function:
39 - W = 0
Solving for W:
W = 39
Since W represents the width, it cannot be negative or greater than 39 (since it corresponds to the length of the other side).
So, the width that maximizes the area is W = 39. We can substitute this value back into the perimeter equation to find the length, L:
2L + 39 = 78
2L = 78 - 39
2L = 39
L = 19.5
Therefore, the length of the rectangular lot that maximizes the area is L = 19.5 feet, and the width is W = 39 feet.