A rancher has 900 meters of fence to enclose a rectangular corral. The corral is to be divided into four subcorrals. What are the overall dimensions of the large enclosure that yield the maximum area?
let the width of each smaller enclosure be x
let the length of each smaller enclosure be y
So 8x + 2y = 900
y = 450 - 4x
Large area = A = 4xy
= 4x(450-4x)
= 1800x - 16x^2
dA/dx = 1800 - 32x
= 0 for a max of A
32x = 1800
x = 56.25 and y = 450 - 4(56.25) = 225
So the large enclosure is 4x by y
or 225 by 225
(no surprise that the large field would be a square, usually this type of question has one side against a barn or a river)
To find the overall dimensions of the large enclosure that yield the maximum area, we need to follow these steps:
Step 1: Understand the problem
We are given that the total length of fence available is 900 meters, and we need to divide the enclosure into four subcorrals. We have to determine the dimensions of the large enclosure that will maximize the area.
Step 2: Establish the variables
Let's assume the length of the large enclosure is "L" meters and the width is "W" meters. We are looking for the values of L and W that maximize the area.
Step 3: Set up the equations
We know that the perimeter of the large enclosure is equal to the sum of the perimeters of the four subcorrals. The perimeter can be calculated by adding the length and width together and multiplying by 2:
P = 2L + 2W
We also know that the total fence available is 900 meters:
2L + 2W = 900
Step 4: Solve the equations
We have two equations:
2L + 2W = 900 --> L + W = 450 (dividing both sides by 2)
Now, we need to express one variable in terms of the other, so let's solve for L:
L = 450 - W
Step 5: Substitute and express the area in terms of one variable
The area of the large enclosure can be calculated by multiplying the length and width together:
A = L * W
Substituting the expression for L from step 4:
A = (450 - W) * W
Step 6: Maximize the area
To find the maximum area, we need to find the value of W that maximizes the equation for A. We can accomplish this by taking the derivative of A with respect to W and setting it equal to 0.
dA/dW = 0 (to find the maximum, the derivative should be zero)
Step 7: Solve the derivative equation
Let's take the derivative of A with respect to W:
dA/dW = (450 - 2W)
Setting it equal to 0:
450 - 2W = 0
Step 8: Find the value of W
Solving for W:
-2W = -450
W = 225 meters
Step 9: Find the value of L
By substituting the value of W back into the equation from step 4:
L = 450 - W
L = 450 - 225
L = 225 meters
Step 10: Check for accuracy
To confirm that our solution yields the maximum area, we can take the second derivative of A with respect to W. If it's positive, then we have found a maximum.
Taking the second derivative of A with respect to W:
d^2A/dW^2 = -2
Since the second derivative is negative, we have found a maximum.
Step 11: Result
The overall dimensions of the large enclosure that yield the maximum area are a length (L) of 225 meters and a width (W) of 225 meters.
To find the overall dimensions of the large enclosure that yield the maximum area, we can follow these steps:
Step 1: Let's assume the length of the large enclosure is 'L' meters and the width is 'W' meters.
Step 2: Since the corral is divided into four subcorrals, each subcorral will have a length of L/2 and a width of W/2.
Step 3: The perimeter of the corral is made up of the lengths of all four sides, so it can be expressed as: 2(L/2) + 2(W/2) = L + W.
Step 4: According to the problem statement, the rancher has 900 meters of fence available, so the perimeter of the corral is 900 meters. Therefore, we can say: L + W = 900.
Step 5: We need to express the area of the corral in terms of L and W. The area of a rectangle is given by: A = L * W.
Step 6: We want to find the maximum area, so we can express the area in terms of a single variable. Using the equation from step 4, we can write: W = 900 - L.
Step 7: Substitute the value of W from step 6 into the formula for area: A = L * (900 - L) = 900L - L^2.
Step 8: To find the maximum area, we need to maximize the function A = 900L - L^2. We can do this by finding the vertex of the function. The vertex of a quadratic function in the form of f(x) = ax^2 + bx + c can be found using the formula: x = -b / (2a).
Step 9: In our case, a = -1, b = 900, and c = 0. Plugging these values into the formula, we get: L = -900 / (2*(-1)) = 450.
Step 10: Now that we have the value of L, we can substitute it back into the equation from step 4 to find the value of W: W = 900 - L = 900 - 450 = 450.
Step 11: The overall dimensions of the large enclosure that yield the maximum area are L = 450 meters and W = 450 meters.
Therefore, the large enclosure with dimensions 450 meters by 450 meters will yield the maximum area.