Answer: the largest area that can be enclosed with 64 feet of fencing along the broad side of the barn is 512 square feet.
Step-by-step explanation:
To find the largest area that can be enclosed with 64 feet of fencing, we can use the formula for the area of a rectangle. Let's assume the length of the rectangle is L and the width is W.
Since the farmer plans to enclose the plot along the broad side of his barn, two sides of the rectangle will be equal to the width, W, and the other two sides will be equal to the length plus the width (since they will be connected to the barn).
The perimeter of the rectangle can be expressed as:
2W + L + W = 64
Simplifying this equation, we get:
2W + L + W = 64
2W + L = 64 - W
2W = 64 - L
From this equation, we can see that the length, L, is equal to 64 minus twice the width, W.
Now, let's substitute this value of L into the formula for the area of a rectangle:
Area = L * W
Substituting L = 64 - 2W, we have:
Area = (64 - 2W) * W
To find the largest area, we need to maximize this function. We can do this by finding the maximum value of the quadratic equation.
By graphing the equation or using calculus, we find that the maximum area occurs when W = 16 feet.
Plugging this value back into the equation for the area, we get:
Area = (64 - 2(16)) * 16
Area = 32 * 16
Area = 512 square feet
Therefore, the largest area that can be enclosed with 64 feet of fencing along the broad side of the barn is 512 square feet.