a rancher has 4000 feet of fencing for constructing a rectangular corral. one side of the corral will be formed by a barn and requires no fence. three exterior fences and 2 interior fences partition the corral into 3 rectangular regions. what dimension of the corral maximize the enclosed area? what is the area of that corral?
Let x be the width of the corral parallel to the barn and y be the length perpendicular to the barn. The total length of the fencing is given by:
3y + 2x = 4000
We want to maximize the total area A = xy. To do this, we first need to express either x or y in terms of the other variable. From the equation above, we can solve for x:
x = (4000 - 3y)/2
Now, substitute this expression for x into the area equation:
A = y * (4000 - 3y)/2
To maximize the area, we can find the maximum of this function with respect to y by taking the derivative of A with respect to y and setting it equal to 0:
dA/dy = (4000 - 6y)/2
Setting the derivative equal to 0:
0 = (4000 - 6y)/2
0 = 4000 - 6y
Now, solve for y:
6y = 4000
y = 2000/3 ≈ 666.67 feet
Substitute the value of y back into the expression for x:
x = (4000 - 3*666.67)/2 ≈ 1000 ft
Hence, the dimensions of the corral that maximize the enclosed area are approximately 1000 feet by 666.67 feet.
To calculate the maximum area, multiply:
A = 1000 * 666.67 ≈ 667,670 square feet
To maximize the enclosed area of the corral, we need to determine the dimensions of the corral. Let's go step-by-step to solve the problem.
Step 1: Let's assume the width of the corral is x.
Step 2: Since one side of the corral is formed by a barn and requires no fence, the length of the corral will be 4000 - 2x (since there are three exterior fences).
Step 3: The area of the corral is given by A = length × width.
A = (4000 - 2x) × x
A = 4000x - 2x^2
Step 4: To find the maximum area, we need to take the derivative of A with respect to x and set it equal to zero.
dA/dx = 4000 - 4x = 0
4000 = 4x
x = 1000
Step 5: Substituting the value of x back into the equation to find the length:
length = 4000 - 2x = 4000 - 2(1000) = 2000
Therefore, the dimensions of the corral that maximize the enclosed area are 1000 feet (width) by 2000 feet (length).
Step 6: Now, let's calculate the area of the corral using the dimensions obtained:
Area = length × width
Area = 2000 × 1000
Area = 2,000,000 square feet
So, the area of the corral that maximizes the enclosed area is 2,000,000 square feet.
To find the dimensions of the corral that maximize the enclosed area, we can start by drawing a diagram of the corral and labeling the given information.
Let's assume the outer length of the corral is "L" and the outer width is "W". We know that there are three exterior fences and two interior fences, which means there are four fence segments in total. Since one side of the corral will be formed by a barn and requires no fence, we have three fence segments left to enclose the corral.
Looking at the diagram, we can see that the length of the corral, L, will be the sum of two fence segments and the width of the corral, W, will be the sum of the remaining two fence segments.
Since we have 4000 feet of fencing available, we can write the equation:
2L + 2W = 4000
Rearranging the equation, we get:
2L = 4000 - 2W
L = 2000 - W/2
Now, let's express the area of the corral in terms of L and W. The area, A, will be the product of length and width:
A = L * W
Substituting the value of L from the earlier equation, we get:
A = (2000 - W/2) * W
To maximize the area, we have to find the value of W that maximizes the equation A = (2000 - W/2) * W. We can do this by using calculus or by simplifying and analyzing the equation.
Simplifying the equation, we get:
A = 2000W - (W^2)/2
To find the value of W that maximizes the area, we take the derivative of A with respect to W and set it equal to zero:
dA/dW = 2000 - W = 0
Solving this equation, we find that W = 2000 feet.
Plugging this value of W back into the equation for L:
L = 2000 - W/2
L = 2000 - (2000/2)
L = 2000 - 1000
L = 1000 feet
Therefore, the dimensions of the corral that maximize the enclosed area are a width of 2000 feet and a length of 1000 feet.
To find the area of the corral, we substitute the values of L and W into the area equation:
A = L * W
A = 1000 * 2000
A = 2000000 square feet
Hence, the area of the corral with the maximum enclosed area is 2,000,000 square feet.