Differentiation - cylinder with hemisphere, SA and volume

[megan-18]

To reduce heat loss, the surface area of a hot-water tank must be kept to a minimum. If such a tank is 125 liters in capacity, and can be approximated by a cylinder in shape with a hemispherical end cap; calculate the radius and overall height for minimum heat loss.

Cant seem to get an answer that makes sense. Any ideas?

 

[Deleted member 4993]

To reduce heat loss, the surface area of a hot-water tank must be kept to a minimum. If such a tank is 125 liters in capacity, and can be approximated by a cylinder in shape with a hemispherical end cap; calculate the radius and overall height for minimum heat loss. Cant seem to get an answer that makes sense. Any ideas?

Please show us what you have tried and exactly where you are stuck.

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[HallsofIvy]

The volume of a cylinder, of radius r and height h, is given by $\displaystyle \pi r^2h$ and its surface area is the circumference, $\displaystyle 2\pi r$ times its height, $\displaystyle 2\pi rh$. Here we should include the bottom, a disk of radius r, $\displaystyle pi r^2$ (but not the top since the top is the hemisphere).

The volume of a sphere, of radius r, is $\displaystyle (4/3)\pi r^3$ and its surface area is $\displaystyle 4\pi r^2$ so the volume of the hemispherical top is $\displaystyle (2/3)\pi r^3$ and the surface area is $\displaystyle 2\pi r^2$.

So the total volume of this tank is $\displaystyle \pi r^2h+ (2/3)\pi r^3$ and the surface area is $\displaystyle 2\pi rh+ 3\pi r^2$. Now, what methods do you know for finding maxima and minima of such functions- like setting the partial derivatives equal to 0?