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
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.
To find the dimensions of the field of the largest area that can be fenced with the given amount of fencing, we need to maximize the area of the rectangle.
Let's assume the length of the rectangular field is x feet.
Since the field borders a straight river, we only need to fence the top, bottom, and one side of the rectangle (the side opposite to the river). This means that for a rectangle of length x, we need to use 2x feet of fencing for the top and bottom, and x feet for the side fence. Therefore, the total fencing used is 2x + x = 3x.
Since the farmer has only 2400ft of fencing, we can write the equation:
3x = 2400
Solving for x:
x = 2400 / 3
x = 800
So, the length of the rectangular field is 800ft.
Since the length is 800ft and the river does not require fencing, the width of the rectangular field is not specified. We can assume the width is y feet.
Now, the total area of the field is given by:
Area = length × width
Area = x × y
Area = 800 × y
Area = 800y
To find the maximum area, we need to find the value of y that maximizes the area. Since the width doesn't depend on the amount of fencing available, we can just consider the function that models the area in terms of the width.
Therefore, the function that models the area of the field in terms of its width (y) is:
A(y) = 800y
To find the maximum area, we can take the derivative of A(y) with respect to y and set it equal to zero:
A'(y) = 800
Setting A'(y) equal to zero:
800 = 0
Since the derivative is a constant, it will always be equal to zero. Therefore, we cannot find a maximum value using this method.
Hence, the dimensions of the field of the largest area that can be fenced with 2400ft of fencing are:
Length: 800ft
Width: undetermined
Please note that without further information or constraints on the problem, the width of the field could be infinitely variable and hence, no specific value can be determined for it.
To solve this problem, we can start by drawing several diagrams to visualize the situation. Let's label the width of the rectangular field as "w" and the length as "l". Since the field borders a straight river, we can conclude that the length will be parallel to the river, and the width will be perpendicular to it.
Let's consider the fencing requirements. The farmer has 2400ft of fencing, and he doesn't need to fence the side along the river. This means that the total length of the fencing will be equal to the sum of the width and twice the length of the field:
2w + l = 2400
To derive the equation for the area of the field in terms of one side, we can use the formula for area:
Area = length × width
Substitute the value of the length from the first equation into the area formula:
Area = l × (2400 - 2w - l) = l(2400 - l - 2w) = 2400l - l^2 - 2wl
Now, we can find the values for the dimensions of the field that maximize the area. To do this, we will differentiate the area function with respect to either l or w and set it equal to zero to find critical points. However, since the equation is given in terms of l as a function of w, we will differentiate the area with respect to l:
d(Area)/dl = 2400 - 2l - 2w
Setting the first derivative equal to zero, we have:
2400 - 2l - 2w = 0
Simplifying this equation, we can solve for the width in terms of the length:
w = (2400 - 2l)/2 = 1200 - l
Substituting this value of w back into our expression for the area, we get:
Area = l(2400 - l - 2(1200 - l)) = l(2400 - l - 2400 + 2l) = l(3l - 2400)
Now, we can find the critical points by taking the derivative of the area function with respect to l and setting it equal to zero:
d(Area)/dl = 3l - 2400 = 0
Solving this equation, we find:
3l = 2400
l = 800 ft
Now, substitute this value of l back into the equation for w:
w = 1200 - l = 1200 - 800 = 400 ft
Therefore, the dimensions of the rectangular field with the largest area are 800 ft (length) and 400 ft (width).