A rancher has 360 feet of fencing to enclose two adjacent rectangular corrals. The wall separating the two corrals consists of a single fence. them. What dimensions for one of the individual corrals will produce the largest total area?
Your answer is _____ft
What is the maximum are for the two corrals combined?
Your answer is______ square ft.
To find the dimensions that produce the largest total area, we need to maximize the combined area of the two corrals.
Let's assume that one corral has a length of L feet and a width of W feet.
Since there are two corrals, the wall separating them will have a length of L feet.
The perimeter of one corral will be:
P = 2L + 2W
And the perimeter of both corrals combined will be:
2P = 2(2L + 2W) = 4L + 4W
We know that the total length of fencing available is 360 feet, so:
4L + 4W = 360
Now, let's solve this equation for L in terms of W:
4L = 360 - 4W
L = (360 - 4W)/4
L = 90 - W
The area of one corral will be:
A = L * W = (90 - W) * W = 90W - W^2
To find the maximum area for the two corrals combined, we need to find the maximum value of A. We can do this by finding the vertex of the quadratic equation.
The vertex of a quadratic equation in the form of A(x - h)^2 + k is (h, k).
In this case, our quadratic equation is:
A = 90W - W^2
To put it in the form of A(x - h)^2 + k, we need to complete the square:
A = -(W^2 - 90W)
Completing the square:
A = -(W^2 - 90W + (90/2)^2 - (90/2)^2)
A = -(W^2 - 90W + 2025 - 2025)
A = -(W^2 - 90W + 2025) + 2025
A = -(W - 45)^2 + 2025
So, the vertex is (45, 2025).
Therefore, the dimensions for one of the individual corrals that will produce the largest total area are:
Length (L) = 45 feet
Width (W) = 45 feet
The maximum area for the two corrals combined is:
2A = 2(2025) = 4050 square feet.
To find the dimensions that produce the largest total area, we can use the concept of optimization. Let's start by understanding the problem and the given information.
We have 360 feet of fencing, which will be used to enclose two adjacent rectangular corrals. The wall separating the two corrals consists of a single fence. We need to determine the dimensions of one of the individual corrals that will give us the largest total area.
To solve this problem, we'll follow these steps:
1. Define the variables: Let's assume the length of one corral is x feet. Since the two adjacent corrals share a fence, the width of each corral can be represented by (360 - 3x)/2, where 3x represents the length of two corral sides and the single fence separating the corrals.
2. Determine the formula for the total area: The total area (A) of two adjacent corrals can be calculated by multiplying the length and width of each corral and then adding the two areas. Therefore, A = x * (360 - 3x)/2 + x * (360 - 3x)/2.
3. Simplify the formula: Combine like terms and distribute the x values to simplify the area equation. A = (x * (360 - 3x))/2 + (x * (360 - 3x))/2.
4. Maximize the total area: To maximize the total area, we can take the derivative of the equation with respect to x and solve for x. Then substitute the value of x into the area equation to find the maximum area.
Differentiating the equation A = (x * (360 - 3x))/2 + (x * (360 - 3x))/2 with respect to x, we get dA/dx = (360 - 6x)/2 + (360 - 6x)/2.
Setting dA/dx equal to zero and solving for x gives us:
(360 - 6x)/2 + (360 - 6x)/2 = 0
720 - 12x = 0
12x = 720
x = 60.
5. Substitute x back into the area equation: Plugging the value of x back into the total area equation, we get:
A = 60 * (360 - 3*60)/2 + 60 * (360 - 3*60)/2
A = 60 * 180/2 + 60 * 180/2
A = 9,000 + 9,000
A = 18,000 square feet.
So, the required answer is:
The dimensions for one of the individual corrals that produce the largest total area are 60 ft by (360 - 3*60)/2 ft.
The maximum combined area for the two corrals is 18,000 square feet.
divide the fence equally into lengths and widths.
360/2 = 180 for each
There are 4 lengths and 3 widths, so
180/4 = 45
180/3 = 60
each corral is 45x60