A horticulturalist is building a fence around a rectangular garden using the side of the building for one side of the enclosure. she has 81 feet fencing. what should be the dimensions of the enclosure be so that she can maximize the garden's area?
If the dimensions of the garden are x and y, with the building along side x, then
x+2y = 81
So, the area (xy) is
(81-2y)y = 81y - 2y^2
This is a parabola, with vertex at y=20.25
Now you can figure the other dimension.
I suspect you meant to say 80 ft of fencing, but only you can decide that.
;kn;n
To maximize the garden's area, we need to find the dimensions of the enclosure that will give us the largest possible rectangle.
Let's assume that the length of the garden is x feet. Since the garden is rectangular and the building is used for one side, the width of the garden will be a constant value.
The fencing required for the length of the garden will be 2x feet (as there are two sides of length x), and the fencing required for the width of the garden will be y feet.
Given that the total fencing available is 81 feet, we can create an equation:
2x + y = 81
Now, let's solve this equation for y:
y = 81 - 2x
The area of a rectangle is given by the formula A = length * width. In this case, the length is x, and the width is y.
So, the area of the garden is:
A = x * y
Substituting the value of y from the previous equation:
A = x * (81 - 2x)
To find the maximum area, we need to take the derivative of A with respect to x and set it equal to zero. This will give us the critical point where the area is maximum.
dA/dx = 0
Let's differentiate A with respect to x:
dA/dx = 81 - 4x
Setting it equal to zero:
81 - 4x = 0
Solving for x:
4x = 81
x = 81/4
x = 20.25 feet
Now, substitute this value of x back into the equation for y:
y = 81 - 2x
y = 81 - 2(20.25)
y = 81 - 40.5
y = 40.5 feet
Therefore, to maximize the garden's area with 81 feet of fencing, the dimensions of the enclosure should be approximately 20.25 feet by 40.5 feet.