SOLUTION: The figure shows a right circular cylinder with radius r and height h. The cylinder is OPEN (no top, just the red bottom). We know the volume of the cylinder is 470 cubic feet. We

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Question 642700: The figure shows a right circular cylinder with radius r and height h. The cylinder is OPEN (no top, just the red bottom). We know the volume of the cylinder is 470 cubic feet. We model the total area of the material needed to construct the cylinder: a rectangle to form the lateral surface of the cylinder and a 2r x 2r square (gray and red) from which to cut out the (red) base.
Express the total area of material necessary to build the cylinder, including the waste from cutting out the bottom, as a function of the radius r.
Answer by KMST(5398) About Me  (Show Source):
You can put this solution on YOUR website!
The area of the base of a cylinder with radius r feet is
B=pi%2Ar%5E2.
The volume of a cylinder with that base and height h feet is
B%2Ah=pi%2Ar%5E2%2Ah.
If that volume is 470 cubic feet, then pi%2Ar%5E2%2Ah=470.
We can solve that equation for h,
pi%2Ar%5E2%2Ah=470 --> h=470%2F%28pi%2Ar%5E2%29
The square of fabric 2r feet by 2r needed to cut the base has an area (in square feet) of
2r%2A2r=4r%5E2
The rectangle of fabric needed to cover the lateral surface of the cylinder is as long as the circumference of the base,
2pi%2Ar feet, and h feet high.
Its area is 2pi%2Ar%2Ah.
Substituting the expression for h that we found above, h=470%2F%28pi%2Ar%5E2%29,
the area (in square feet) of the rectangle of fabric to cover the lateral surface of the cylinder is
2pi%2Ar%2A%28470%2F%28pi%2Ar%5E2%29%29=2pi%2Ar%2A470%2F%28pi%2Ar%5E2%29%29=940%28pi%2Fpi%29%28r%2Fr%5E2%29=940%2Fr
Adding to that the surface area of square of fabric needed to cut the base, the total amount of fabric needed (in square feet) is
940%2Fr%2B4r%5E2