Calculus cylinder with hemisphere

[bobsmith]

hi i need help with a calculus question

To reduce heat loss, the surface area of a hot-water tank must be kept to a minimum. If such a tank is 125 litters 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.


i have this done so far am lost in how to simplfily the terms


The volume is the sum of the volume of a hemisphere and a cylinder.
V= 2/3 ?r^3+?r^2 h
The surface area is
S=2?rh+?r^2+ 2?r^2 = 2?rh+3?r^2
Isolate h in the Volume equation.
V= ?r^3+?r^2 h



h=(V- 2/3 ?r^2 )/(?r^2 )

Substitute for h into the Surface area equation.
S=2?r^2 (V- 2/3 ?r^2 )/(?r^2 )+3?r^2

 

[pitoten]

You made an error solving for h in the Volume equation.

You also made a second error when you rewrote the Surface area equation.

Otherwise you are headed in exactly the right direction. Find the correct equation for surface area in terms of r, which would be
S(r). Now find dS/dr and find where it is equal to zero (before you do this, you should simplify S(r) as much as possible once you find it). I believe this is your answer for r; then find h using the Volume equation.

Remember: liters are defined as cubic decimeters, so make sure the units of your answer for r are correct!

 

[bobsmith]

thing is i haven't a clue how to do this

i get very muddled when the equations go in to factions

i 125 litres will make 125000 cm3

 

[galactus]

The volume of the hemispherical part is \(\displaystyle V=\frac{2}{3}{\pi}r^{3}\)

The volume of the cylindrical part is \(\displaystyle V={\pi}r^{2}h\)

Ttoal volume \(\displaystyle 125=\frac{2}{3}{\pi}r^{3}+{\pi}r^{2}h\)................[1]

The surface area of the hemisphere is \(\displaystyle S=2{\pi}r^{2}\)

surface area of cylindrical part is:

\(\displaystyle S=\underbrace{2{\pi}rh}_{\text{body of cylinder}}+\overbrace{{\pi}r^{2}}^{\text{bottom}}\)

Total surface area is: \(\displaystyle 2{\pi}r^{2}+2{\pi}rh+{\pi}r^{2}\)..............[2]

Solve [1] for h and sub into [2]

This gives us \(\displaystyle S=\frac{5({\pi}r^{3}+150)}{3r}\)

Now, differentiate this, set to 0 and solve for r. The height and surface area follow by substitution.