John wants to build a corral next to his barn.He has 300 feet of fencing to enclose three sides of his rectangular yard.
a. What is the largest area that can be enclosed?
b. What dimensions will result in the largest yard?
75 * 150?
50 * 200?
10 * 280?
75 * 150
To find the largest area that can be enclosed, we need to determine the dimensions that will maximize the area of the rectangular yard.
Let's start by understanding the given information. We know that John has 300 feet of fencing to enclose three sides of his yard. This means that the total length of these three sides combined is 300 feet.
Now, let's assume that the width of the yard is x feet. Since the yard has three sides fenced, we can deduce that the length of the yard is (300 - 2x) feet, considering that the remaining fencing is used for the length.
The area of a rectangle is found by multiplying the length and width. In this case, the area A is given by:
A = width * length
A = x * (300 - 2x)
A = 300x - 2x^2
To find the largest area, we need to maximize the equation A = 300x - 2x^2. To do this, we can take the derivative of A with respect to x and set it equal to zero.
dA/dx = 300 - 4x
Setting dA/dx = 0 gives us:
300 - 4x = 0
4x = 300
x = 75
So, the width of the yard that will result in the largest area is 75 feet.
To find the length of the yard, we substitute this value of x back into the equation of the length:
length = 300 - 2x
length = 300 - 2*75
length = 150
Therefore, the dimensions that will result in the largest yard are a width of 75 feet and a length of 150 feet.
Finally, to find the largest area that can be enclosed:
Area = width * length
Area = 75 * 150
Area = 11,250 square feet.