You have 600 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?
area= LW
but 2W+L=600
L= 600-2W
area= (600-2W)(w)
well, roots are w=0, and w=300 that is where area is zero. Since a parabola is described here, the max will occur halfway, or w=150. Solve for Area when w=150
Plot Area vs W and see if this is so. Use your graphing calculator.
I already know that the answer is 300 for the length and 150 for the width, giving a sq ft of 45,000. The only problem is i do not know how to get that answer.
@bobpursley i got it now. thanks a bunch =)
To find the length and width of the plot that will maximize the area, we can use calculus. Let's start by assigning variables to the length and width of the plot.
Let's assume the length of the plot is L, and the width is W. Since the side along the river does not need to be fenced, we have two sides with length L and two sides with length W.
The perimeter of the fenced region is given as 600 feet, so we can write the equation:
2L + W = 600
To make it easier, let's rewrite it as:
W = 600 - 2L
Now, we can express the area of the plot, A, in terms of L and W:
A = L * W
Substituting the value of W from the previous equation, we have:
A = L * (600 - 2L)
To find the maximum area, we need to find the critical points of the function A(L). We can do this by finding the derivative of A with respect to L, and setting it equal to zero:
dA/dL = 600 - 4L = 0
Solving for L, we get:
600 - 4L = 0
4L = 600
L = 150
So, the length of the plot that will maximize the area is 150 feet. To find the width, we can substitute this value back into the equation we derived earlier:
W = 600 - 2L
W = 600 - 2(150)
W = 300
Therefore, the width of the plot is 300 feet.
To find the largest area that can be enclosed, we can now substitute the values of L and W into the area equation:
A = L * W
A = 150 * 300
A = 45,000 square feet
Hence, the largest area that can be enclosed is 45,000 square feet.