Optimization question - optimal conical container

[miss dhia]

Design the optimal conical container that has a cover and has walls of negligible thickness. The container is to hold 0.5 m^3. Design it so that the areas of its base and sides are minimized.

information :
1) areas of the sides = (pi) x r x s
2) areas of the base = (pi) x (r^2)
3)volume of the cone = (1/3) x (pi) x (r^2) x h

 

[tiny-tim]

welcome to pf!

hi miss dhia! welcome to pf!

(have a pi: π and try using the X² icon just above the Reply box )

1) areas of the sides = (pi) x r x s
2) areas of the base = (pi) x (r^2)
3)volume of the cone = (1/3) x (pi) x (r^2) x h

what is s ?

 

[miss dhia]

hye tiny-tim! thanks for your concern on this question..

for your information, i am attaching the picture of the conical container..may help you to solve this question..thanks in advance! ;)

 

[miss dhia]

ooppss tiny-tim, i am forgotten to tell you what is r, h, and s actually.

actually,
s = side of the conical container (shown in picture).
r = radius of conical container (shown in picture).
h = height of the conical container (shown in picture)

 

[chiro]

Design the optimal conical container that has a cover and has walls of negligible thickness. The container is to hold 0.5 m^3. Design it so that the areas of its base and sides are minimized.

information :
1) areas of the sides = (pi) x r x s
2) areas of the base = (pi) x (r^2)
3)volume of the cone = (1/3) x (pi) x (r^2) x h

Using your information can you get an expression for the total area in terms of one other variable?

 

[tiny-tim]

hi miss dhia!

what is s ?

s = side of the conical container (shown in picture).

actually, i meant what is s in terms of r and h?

you need to convert it so that all three areas are written in r and h, and you can easily add them

 

[miss dhia]

hye tiny-tim!

s = \sqrt{r^{} + h^{}}

i don't know how to put the "surd". sorry ok. :)

in the above equation, i do mean, s = surd(r^2 + h^2).

 

[tiny-tim]

hye miss dhia!

yup, s = √(r² + h²) …

ok, now write out the formulas for A and for V,

and see if you can find a relationship between them