To find the dimensions of the garden that will give the greatest possible area, we need to use the formula for the perimeter of a rectangle: P = 2L + 2W, where P is the perimeter, L is the length, and W is the width.
Given that the Wilsons have 12 yards of fence for their garden, we can set up the equation:
2L + 2W = 12
Simplify the equation to:
L + W = 6
L = 6 - W
Now, to find the greatest possible area, we can use the formula for the area of a rectangle: A = L * W.
Substitute the equation for L into the formula for the area:
A = (6 - W) * W
A = 6W - W^2
To find the maximum area, we need to find the vertex of the parabola A = -W^2 + 6W.
The vertex of a parabola in the form y = ax^2 + bx + c is given by x = -b / 2a. In this case, a = -1, and b = 6.
x = -6 / 2(-1)
x = -6 / -2
x = 3
Therefore, the width that gives the greatest possible area is 3 yards. To find the length, we can use the equation L + W = 6:
L + 3 = 6
L = 6 - 3
L = 3
Therefore, the length of the garden is 3 yards, and the width of the garden is 3 yards, which will give the greatest possible area of 9 square yards.
1. The Wilsons have 12 yards of fence for their garden. What are the length and width of the garden if it has the greatest possible area?
1 answer