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?
To find the largest area that can be enclosed and the dimensions that will result in the largest yard, we can use some algebraic reasoning.
a. To find the largest area that can be enclosed, let's assume that two of the sides of the rectangle have lengths x and y, and the third side is the barn. We know that the total length of the three sides is 300 feet.
So, the equation becomes: 2x + y = 300.
Now, we want to find the area of the rectangle, which is given by length times width. In this case, it is xy.
We can solve for either x or y in terms of the other variable and substitute into the area equation to get a single variable equation in terms of one variable only. Let's solve for y in terms of x:
2x + y = 300
=> y = 300 - 2x
Now, substitute y in the area equation:
Area (A) = x(300 - 2x) = 300x - 2x^2
To determine the largest possible area, we need to find the maximum value of A. 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 the derivative equal to zero and solving for x:
300 - 4x = 0
=> 4x = 300
=> x = 75
To find the corresponding y-value, substitute x back into the equation for y:
y = 300 - 2x
=> y = 300 - 2(75)
=> y = 150
Therefore, the dimensions that result in the largest yard are 75 feet by 150 feet.
b. The dimensions that result in the largest yard are 75 feet by 150 feet.