SOLUTION: A gardening company is designing a rectangular compost container that will be twice as tall as it is wide and must hold 18 ft3 of composted food scraps. Find the dimensions of the

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Question 1040831: A gardening company is designing a rectangular compost container that will be twice as tall as it is wide and must hold 18 ft3 of composted food scraps. Find the dimensions of the compost container that has minimal surface area. (include the top and bottom of the container)
Found 3 solutions by josgarithmetic, josmiceli, robertb:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
x and 2x rectangular dimensions, and y for the height.

system%282x%5E2%2Ay=18%2CS=2%2A2x%2Ax%2B2xy%2B2%2A2xy%29;
first equation gives y=9%2Fx%5E2

S=4x%5E2%2B2xy%2B4xy
S=4x%5E2%2B2x%289%2Fx%5E2%29%2B4x%289%2Fx%5E2%29
S=4x%5E2%2B18%2Fx%2B36%2Fx
S=%284x%5E3%2B18%2B36%29%2Fx
S=%284x%5E3%2B54%29%2Fx

Quotient Rule
dS%2Fdx=%28x%2A12x%5E2-%284x%5E3%2B54%29%2A1%29%2F%28x%5E2%29
dS%2Fdx=%2812x%5E3-4x%5E3-54%29%2Fx%5E2
-
8x%5E3-54=0
4x%5E3=27
x%5E3=27%2F4
highlight%28x=3%2Froot%283%2C4%29%29

Now also find 2x and y.

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +W+ = the width
+2W+ = the height
Let +L+ = the length
--------------------
given:
+W%2A2W%2AL+=+18+ ft3
+2L%2AW%5E2+=+18+
+L%2AW%5E2+=+9+
+L+=+9+%2F+%28W%5E2%29+
---------------------
Let +S+ = the inside surface area
+S+=+2%2AW%2A2W+%2B+2%2AL%2A2W+
+S+=+4W%5E2+%2B+L%2A4W+
+S+=+4W%5E2+%2B+%28+9%2F%28W%5E2%29%29%2A4W+
+S+=+4W%5E2+%2B+36%2FW+
Is this in a calculus course? If so take the
derivative of +S+ with respect to +W+
------------------------------------
+s%5B1%5D+=+8W+-+72%2F%28W%5E2%29+
+8W+-+72%2F%28W%5E2%29+=+0+
+8W+=+72%2F%28W%5E2%29+
+W%5E3+=+9+
+W+=+root%28+3%2C+9+%29+
+W+=+2.08008+
height = +2W+=+4.16016+
length = +L+=+9+%2F+%28W%5E2%29+
length = +9+%2F+4.32673+
length = +2.08008+
------------------------
check:
+2.08008%2A2.08008%2A4.16016+=+18+
+17.9999+=+18+
--------------------
Here's a plot of the surface area function,
+S+ and +W+
--------------------
+graph%28+400%2C+400%2C+-10%2C+10%2C+-100%2C+100%2C+4x%5E2+%2B+36%2Fx+%29+
Hope I got it -definitely get another opinion on this!!!

Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
If x = width, l = length, then h = 2x.
==> x%5E2%2Al+=+9, and A%5Bs%5D+=+6lx%2B4x%5E2, where A%5Bs%5D is the total surface area.
There is be an absolute minimum at x+=+3%2Froot%283%2C4%29+=+highlight%28%283%2F2%29%2Aroot%283%2C2%29%29 feet.
==> highlight%28l+=+2root%283%2C2%29%29 feet, and height = highlight%283root%283%2C2%29%29 feet.