Asked by Ray
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?
What is the minimum total material needed to build such an enclosure?
Answers
Answered by
Reiny
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
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
Answered by
Damon
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?
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?
Answered by
Steve
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
So, you know right off that 2y=3x.
y(2y/3) = 225
y^2 = 675/2
y = 18.37
x = 12.23
Answered by
Ray
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?????
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?????