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
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.
To find the dimensions of the field with the largest area, we can use calculus. Let's assume the width of the field is x feet.
Since the fencing only needs to be placed on three sides (the river acts as a natural boundary for the fourth side), the total length of fencing required is 2400 ft. This means that the sum of all the sides (including the river side) should be equal to 2400 ft.
The equation representing the total length of fencing is:
x + 2y = 2400
To find the area of the field, we multiply the width (x) by the length (y):
Area = x * y
Now, we need to solve the equation for y in terms of x:
y = (2400 - x) / 2
Substitute this value of y into the area equation:
Area = x * (2400 - x) / 2
To find the dimensions that yield the largest area, we can take the derivative of the area function with respect to x, set it equal to zero, and solve for x.
d(Area) / dx = (2400 - 2x) / 2
Setting d(Area) / dx equal to zero:
(2400 - 2x) / 2 = 0
Simplifying:
2400 - 2x = 0
2x = 2400
x = 1200
So, the width of the field with the largest possible area is 1200 ft.
To find the length (y) of the field, we substitute this value of x into the equation for y:
y = (2400 - x) / 2
y = (2400 - 1200) / 2
y = 1200 / 2
y = 600
Therefore, the dimensions of the field with the largest area that can be fenced off are 1200 ft by 600 ft.
To solve this problem, we can use calculus to find the dimensions of the field with the largest area.
Let's assume the length of the field is x feet. Since the field is rectangular, we can also assume that the width is y feet.
We are given that the total amount of fencing the farmer has is 2400ft. The fencing is used for the perimeter of the field, so we can set up the following equation:
Perimeter of the field = 2x + y = 2400
We are also given that the field borders a straight river, so the length of the fence along the river is not needed. This means that we can eliminate one side of the perimeter equation:
2x + y - x = 2400
x + y = 2400
Now, we need to use the equation for the area of a rectangle to find the area of the field, which is given by:
Area of the field = x*y
To find the dimensions of the field with the largest area, we need to maximize this function.
To do that, we need to solve the system of equations:
x + y = 2400
Area of the field = x*y
Solving the first equation for y, we get y = 2400 - x.
Substituting this into the equation for the area of the field, we get:
Area of the field = x*(2400 - x) = 2400x - x^2
We can now find the maximum value of this function using calculus. To do this, we take the derivative of the area function with respect to x, set it equal to zero, and solve for x.
d(Area of the field)/dx = 2400 - 2x = 0
2400 - 2x = 0
2x = 2400
x = 1200
Substituting this value of x back into the equation for the perimeter, we can find the value of y:
x + y = 2400
1200 + y = 2400
y = 1200
Therefore, the dimensions of the field of largest area that the farmer can fence off are 1200 feet by 1200 feet.