Let the length of the field be L feet and the width of the field be W feet. Since the field is rectangular, the perimeter of the field is given by 2L + W feet.
We are given that the farmer has 2400ft of fencing, so we can write the equation:
2L + W = 2400
To maximize the area of the field, we need to find the maximum value of the area. The area of the field is given by A = L * W.
To solve for the dimensions of the field of largest area, we can use the equation 2L + W = 2400 to express W in terms of L:
W = 2400 - 2L
Substituting this value of W in the equation for the area:
A = L * (2400 - 2L)
A = 2400L - 2L^2
To find the maximum area, we need to find the vertex of the quadratic equation A = 2400L - 2L^2. The vertex of a quadratic equation of the form y = ax^2 + bx + c is given by x = -b/2a.
In this case, a = -2, b = 2400, and c = 0. Therefore, the value of L that maximizes the area of the field is L = -2400 / (2 * -2) = 600.
Substituting this value of L back into the equation for W:
W = 2400 - 2(600) = 1200
Therefore, the dimensions of the field of largest area that the farmer can fence off are: L = 600 feet and W = 1200 feet.
The smaller dimension is 600 feet and the larger dimension is 1200 feet.
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
Find a function that models the area of the feild in terms of one of its sides
Use your model to solve the problem and compare with your answer to part (a) maximum area occurs at the following values
Smaller dimension and larger dimension in feet
1 answer
1 answer