Carol has 1100 feet of fencing to fence in a rectangular horse corral
Find a function that models the area A of the corral in terms of the width x of the corral A(x)
Find the dimensions of the rectangle that maximize the area of the corral
Width and length in ft
To find a function that models the area A of the corral in terms of the width x, we can assume that the length of the corral is also x. Therefore, the area of the corral can be represented as:
A(x) = x * x = x^2
To find the dimensions of the rectangle that maximize the area of the corral, we can use the fact that Carol has 1100 feet of fencing. The perimeter of the rectangle will be equal to the total length of fencing, so we have:
2x + 2x = 1100
4x = 1100
x = 275
Therefore, the width of the corral is 275 ft and the length is also 275 ft.
To find a function that models the area A of the corral in terms of the width x, we need to express the length of the corral in terms of the width.
Let's assume the width of the corral is x feet. Since the corral is rectangular, then its length will be (1100 - 2x) feet. This is because we need to subtract the lengths of the two sides that are parallel to the width, which each have a length of x.
So, the area A of the corral can be expressed as the product of the width and length:
A(x) = x * (1100 - 2x)
To find the dimensions of the rectangle that maximize the area of the corral, we need to find the value of x that maximizes the function A(x).
The dimensions that maximize the area occur at the maximum point of the function, which can be found by setting the derivative of A(x) equal to zero and solving for x:
A'(x) = 0
d/dx [x * (1100 - 2x)] = 0
To find the critical point(s), we take the derivative of A(x) and set it equal to zero:
1100 - 4x = 0
Solving for x, we get:
4x = 1100
x = 275
Therefore, the width of the corral that maximizes the area is 275 feet. To find the length, we substitute this value back into the expression for length above:
Length = 1100 - 2x = 1100 - 2(275) = 550 feet
So, the dimensions of the rectangle that maximize the area of the corral are width = 275 feet and length = 550 feet.
To find a function that models the area A of the corral in terms of the width x of the corral, let's break down the problem step by step.
We know that the perimeter of a rectangle is given by the formula: perimeter = 2(length + width). In this case, the perimeter of the corral is given as 1100 feet. So we have the equation:
1100 = 2(length + x)
Let's solve this equation for the length in terms of x:
1100 = 2(length + x)
550 = length + x
length = 550 - x
Now that we have the length in terms of x, we can determine the area of the corral. The area of a rectangle is given by the formula: area = length * width. Substituting the previous equation for length, we have:
A(x) = (550 - x) * x
This is the function that models the area A of the corral in terms of the width x of the corral.
Now let's find the dimensions of the rectangle that maximize the area of the corral. To do this, we need to find the value of x that maximizes the function A(x). We can accomplish this by finding the critical points of the function.
To find the critical points, we differentiate the function A(x) with respect to x and set it equal to zero:
A'(x) = (550 - 2x) * 1 = 0
Simplifying the equation, we have:
550 - 2x = 0
2x = 550
x = 275
To confirm that this is a maximum, we can take the second derivative of A(x). If it is negative, then we have a maximum.
A''(x) = -2 < 0
Since the second derivative is negative, we can conclude that x = 275 is the width that maximizes the area of the corral.
To find the length, we substitute this value of x into our previous equation for length:
length = 550 - x
length = 550 - 275
length = 275
Therefore, the dimensions of the rectangle that maximize the area of the corral are width = 275 ft and length = 275 ft.