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?
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.