Ann wants to plant a rectangular garden next to a straight river. She has 120 feet of fencing and will not fence the side of the garden next to the river. What dimensions will produce the largest garden?
x+2y = 120
A = x y
A = (120-2y)y = 120 y - 2 y^2
2 y^2 -120 y = -A
y^2 -60 y = -A/2
y^2 - 60 y + 900 = -A/2 + 900
(y-30)^2 = -1/2(A -1800)
so y = 30
x = 120 - 60 = 60
sure enough 30*60 = 1800
To determine the dimensions that will produce the largest garden, we need to find the shape with the maximum area given the constraint of 120 feet of fencing.
Let's denote the length of the garden as L and the width as W.
We know that two sides of the garden, which are parallel to the river, do not require fencing. This means that the total length of the fencing will be the two remaining sides plus the width. So, the perimeter of the garden is:
Perimeter = 2L + W
According to the problem, we have 120 feet of fencing. Therefore:
2L + W = 120
Now, let's express W in terms of L:
W = 120 - 2L
To find the area of the garden, we multiply the length and width:
Area = L * W
Substituting the value of W from the equation above:
Area = L * (120 - 2L)
To find the dimensions that produce the largest garden, we need to find the maximum value of the area. We can do this by either finding the vertex of the parabola or by finding the value of L that maximizes the area.
To proceed, we can differentiate the area equation with respect to L and set the derivative equal to zero:
d(Area)/dL = 120 - 4L = 0
Solving for L:
120 - 4L = 0
4L = 120
L = 30
Now that we have the value of L, we can substitute it back into the equation for W:
W = 120 - 2L = 120 - 2(30) = 60
Therefore, the dimensions that will produce the largest garden are 30 feet (length) by 60 feet (width).