Laura has 20 feet of fencing to make a rectangular pen for her dog. What is the largest area that she can fence in?
To find the largest area that Laura can fence in with 20 feet of fencing, she should make a rectangular pen with dimensions that maximize the area.
Let's assume the length and width of the rectangular pen are L and W, respectively. Since the pen is rectangular, it will have two lengths (L) and two widths (W).
To maximize the area, we need to find the dimensions that will give us the largest possible product between L and W.
Since Laura has 20 feet of fencing, the perimeter of the pen would be 20 feet. The perimeter of a rectangle is calculated by adding up all the sides.
Therefore, we have the equation: 2L + 2W = 20.
To maximize the area, we want to isolate one variable (either L or W) and express it in terms of the other. Let's isolate L:
2L = 20 - 2W (subtracting 2W from both sides)
L = 10 - W (dividing both sides by 2)
Now we can express the area of the rectangular pen in terms of W:
Area = L * W
Area = (10 - W) * W
Area = 10W - W^2
To find the maximum area, we take the derivative of the area equation with respect to W, set it to zero, and solve for W. This will give us the width that maximizes the area.
d(Area)/dW = 10 - 2W
Setting the derivative to zero:
10 - 2W = 0
2W = 10
W = 5
So, the width that maximizes the area is 5 feet. Now we can find the length (L):
L = 10 - W
L = 10 - 5
L = 5
Therefore, the length is also 5 feet.
The largest area Laura can fence in is found by multiplying the width and length: Area = L * W = 5 * 5 = 25 square feet.
So, the largest area Laura can fence in is 25 square feet.