Question 347119
A tank is to be constructed that will hold 5.00 X 10^5L when filled. The Shape is to be cylindrical with a hemispherical top. find the dimensions of the container (that has a volume of 500000 liters) so you can calculate the surface area.
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h = height of the cylinder
r = radius of the cylinder and of the hemisphere
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-----Cyl vol + Hemisph vol
{{{Vol = pi*r^2*h + 2*pi*r^3/3 = 500000}}}
{{{500000 = pi*r^2*(h + 2r/3)}}}
{{{500000/(pi*r^2) = h + 2r/3}}}
{{{h = 500000/(pi*r^2) - 2r/3}}}
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{{{SA = 2pi*r*h + 2pir^2}}}
{{{SA = 2pi*r*(500000/(pi*r^2) - 2r/3) + 2pi*r^2}}}
{{{SA = 1000000/r - 4pi*r^2/3 + 2pi*r^2}}}
That's the surface area in terms of r.   No numerical value of SA can be determined without a value of r. (The bottom is not included, btw)
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If you want the value of r that gives the minimum SA for the given shape and volume, that can be determined.