Question 642700
The area of the base of a cylinder with radius {{{r}}} feet is
{{{B=pi*r^2}}}.
The volume of a cylinder with that base and height {{{h}}} feet is
{{{B*h=pi*r^2*h}}}.
If that volume is 470 cubic feet, then {{{pi*r^2*h=470}}}.
We can solve that equation for {{{h}}},
{{{pi*r^2*h=470}}} --> {{{h=470/(pi*r^2)}}}
The square of fabric {{{2r}}} feet by {{{2r}}} needed to cut the base has an area (in square feet) of
{{{2r*2r=4r^2}}}
The rectangle of fabric needed to cover the lateral surface of the cylinder is as long as the circumference of the base,
{{{2pi*r}}} feet, and {{{h}}} feet high.
Its area is {{{2pi*r*h}}}.
Substituting the expression for {{{h}}} that we found above, {{{h=470/(pi*r^2)}}},
the area (in square feet) of the rectangle of fabric to cover the lateral surface of the cylinder is
{{{2pi*r*(470/(pi*r^2))=2pi*r*470/(pi*r^2))}}}={{{940(pi/pi)(r/r^2)=940/r}}}
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/r+4r^2}}}