Question

Owner wants to fence in 1000 ft ^2 of land in a rectangular plot to be used for different types of shrubs. The plot is to be divided into four equal plots with three fences parallel to the same pair of sides. What is the least number of feet of fence needed?

Answer 1

1000/4 = 250 sq feet for each plot

Answer 2

factor 250, to get 2 plausible values for length and width for each plot

then add up the width and length used

Answer 3

Refer to the figure above.

The area of the rectangular piece of land is:
A = xy

The total length of fencing is:
L = 2x + 5y

We know the area of the piece of land is 1000 ft^2, so we can express y in terms of x:
A = xy = 1000
y = 1000/x

Now we plug in y into the equation for the length of fencing to get the length of fencing as a function of x:
L = 2x + 5y
L = 2x + 5(1000/x)
L = 2x + 5000/x
L = 2x + 5000x^(-1)

To find a maximum or minimum, we take the first derivative and set it equal to zero:
dL/dx = 2 + 5000(-1)x^(-2)
dL/dx = 2 - 5000/x^2
2 - 5000/x^2 = 0
2x^2 -5000 = 0
2x^2 = 5000
x^2 = 2500
x = 50 or x = -50
Discard x = -50
x = 50
We now know that x = 50 ft.

To find the total length of fencing, we plug in x = 50 into our expression of the length of fencing in terms of x:
L = 2x + 5000/x
L = 2(50) + 5000/50
L = 100 + 100
L = 200

The minimum length of fencing is 200 ft.