A rectangular fenced enclosure of area 225 square feet is divided half into 2 smaller rectangles.
What is the minimum total material needed to build such an enclosure?
Make a sketch of a rectangle with a line parallel to the width dividing the rectangle into two equal parts.
Let the length of the smaller rectangle be x and its width equal to y.
(the whole rectangle would be 2x by y)
given: 2xy = 225
y = 225/(2x)
length of fence = L
L = 4x + 3y
L = 4x + 675/2x = 4x + 337.5/x
dL/dx = 4 - 337.5/x^2
= 0 for a minimum of L
4 = 337.5/x^2
x^2 = 84.375
x = appr 9.19 ft
y = 18.37 ft
L = 91.86 ft as the minimum
three fences of length x
two fences of length y
A = xy = 225 so x = 225/y
p = total length = 2y + 3 x
we want to minimize p
p = 2 y +3(225/y)
dp/dy = 0 at max or min
dp/dy = 0 = 2 -675/y^2
2 y^2 = 675
y^2 = 337
y = 18.4
x = 225/18.4
so what is 2y+3x?
As always, in similar problems, the fence is evenly divided into lengths and widths.
So, you know right off that 2y=3x.
y(2y/3) = 225
y^2 = 675/2
y = 18.37
x = 12.23
Right!
Which apparently that isn't the right answer! I did pretty much exactly what @Damon and @Steve but my teacher marked it as wrong. What gives?????
To find the minimum total material needed to build the enclosure, we need to determine the dimensions of the smaller rectangles and calculate the perimeter of the fence.
First, let's find the dimensions of the two smaller rectangles. Since the rectangular enclosure is divided in half, it means the area is split equally between the two rectangles. So each rectangle will have an area of 225 square feet ÷ 2 = 112.5 square feet.
Now, we need to find two positive numbers whose product is 112.5 and whose sum is minimized. To do this, we can use the fact that the minimum of a sum occurs when the two numbers are closest together.
Let's start by finding the square root of 112.5. √112.5 ≈ 10.61
Now, we can choose two numbers close to 10.61 that multiply to 112.5. The closest numbers are 10 and 11. So the dimensions of the two rectangles will be 10 feet by 11.25 feet each.
To calculate the minimum total material needed, we need to find the sum of the perimeters of the two rectangles.
The perimeter of a rectangle is given by the formula: P = 2(length + width)
For the first rectangle:
P1 = 2(10 + 11.25) = 2(21.25) = 42.5 feet
For the second rectangle (which has the same dimensions as the first):
P2 = P1 = 42.5 feet
The minimum total material needed to build the enclosure is the sum of the perimeters of the two rectangles:
Total material needed = P1 + P2 = 42.5 feet + 42.5 feet = 85 feet
Therefore, the minimum total material needed to build the enclosure is 85 feet.