Answer:
The dimensions that maximize the area of the garden are 200 feet by 100 feet.
The largest area for the garden would be 20,000 square feet
Explanation:
Let the length of the garden be x feet and the width be y feet. One side of the garden is adjacent to the barn, which is 50 feet long. Since only three sides need to be fenced, we can set up the following equation for the perimeter:
x + 2y = 400
We need to find the dimensions that will maximize the area of the garden, which is given by A = xy. First, we will express y in terms of x using the perimeter equation:
y = (400 - x) / 2
Now, substitute y in the area equation:
A(x) = x(400 - x) / 2
A(x) = (400x - x^2) / 2
To maximize the area, we need to find the critical points of the function A(x). We do this by taking the first derivative and setting it to 0:
A'(x) = (400 - 2x) / 2
A'(x) = 200 - x
Now, set A'(x) to 0 and solve for x:
200 - x = 0
x = 200
Now, substitute x = 200 back into the equation for y:
y = (400 - 200) / 2
y = 200 / 2
y = 100
So, the dimensions that maximize the area of the garden are x = 200 feet (length) and y = 100 feet (width). Now we can find the largest possible area:
A = xy
A = (200)(100)
A = 20,000 square feet
The largest possible area for the garden is 20,000 square feet.