To find the dimensions of the field that will yield the largest area, we can use the method of calculus by maximizing the area function subject to the constraint of the total amount of fencing.
Let's assume the width of the field is x and the length of the field (parallel to the river) is y.
The perimeter of the field will then be: P = 2x + y
Given that the total amount of fencing available is 2400ft, we have the equation:
2x + y = 2400
We want to maximize the area of the field, which is given by: A = xy
Using the constraint equation, we can solve for y and substitute into the area equation:
y = 2400 - 2x
A = x(2400 - 2x)
To find the maximum area, we need to find the critical points of the area function. We differentiate the area function with respect to x and set it equal to zero to find the critical points:
dA/dx = 2400 - 4x = 0
4x = 2400
x = 600
Substituting x = 600 back into the constraint equation, we find the corresponding value of y:
2(600) + y = 2400
1200 + y = 2400
y = 1200
Therefore, the dimensions of the field that will yield the largest area are 600ft (width) and 1200ft (length).
So, the answer is: 600, 1200.
A farmer has 2400ft of fencing and wants to fence off a rectangular field that borders a straight river he does not need a fence along the river what are the dimensions of the feild of largest area that he can fence
Experiment with the problem drawing several diagrams illustrating the situation calculate the area of each configuration and use your results to estimate the dimensions of target possible field enter your answer as a comma separated list
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1 answer