Your grandmother has asked you to help her build a rectangular garden for her roses in her back yard. She has 120 feet of fencing and plans to have one side of the garden against the back of her house so it doesn’t need fencing. What are the dimensions (length and width) of the garden that would give her the maximum possible area for growing her beloved roses?
L + 2 W = 120 ... L = 120 - 2W
area = L * W = (120 - 2W) * W = 120 W - 2 W^2
the maximum is on the axis of symmetry ... -b / 2a
Wmax = -120 / (2 * -2)
substitute back to find Lmax
To find the dimensions that would give your grandmother the maximum possible area for her garden, we need to apply a mathematical concept called optimization.
Let's assume the width of the garden as x feet. Since one side of the garden is against the back of the house, an equal length of fencing is saved. Therefore, the length of the garden will be 120 - 2x feet (subtracting the saved length from the total fencing).
To find the area of the garden, we multiply the length and width:
Area = length * width
Area = (120 - 2x) * x
Area = 120x - 2x²
Now, we want to maximize the area, so we need to find the maximum value of this equation. To do that, we can take the derivative of the equation and set it equal to zero:
d(Area)/dx = 120 - 4x = 0
Solving for x:
120 - 4x = 0
4x = 120
x = 30
So, the width of the garden is 30 feet. The length can be found by substituting the width value into the expression for length:
Length = 120 - 2x
Length = 120 - 2(30)
Length = 120 - 60
Length = 60
Therefore, the dimensions of the garden that would give your grandmother the maximum possible area for growing her roses are 60 feet (length) by 30 feet (width).