The cooling will be dependent on surface area.
The max surface area will cool fastest. It wont take a genius to make a cylinder which has greater surface area than the other two shapes.
Philip forms three equal portions of leftover mashed potatoes into three shapes: a cube, a sphere, and a cylinder. Each portion is heated in the oven to the same uniform temperature.
When he takes the portions out of the oven and leaves them at room temperature, which shape takes the longest time to cool down?
Assume that the mashed potatoes are uniform in composition and that each shape is maintained throughout the process.
2 answers
which has the lowest ratio of surface area to volume?
let volumes all be 1
sphere
v = (4/3) pi r^3 = 1
then r = [ 3/(4pi) ]^1/3
r = .620
a = 4 pi r^2 = 4.836
cube
v = s^3 = 1 so s = 1
a = 6 s^2 = 6 (so cools faster than sphere)
cylinder
LOL, depends on height to radius but
pi r^2 h = 1 so h = 1/(pi r^2)
and a = 4 pi r^2 + 2 pi r h
so
a = 4 pi r^2 + 2 pi r * 1/(pi r^2)
a = 4 pi r^2 + 1/r
look for max or min area for this volume
0 = 8 pi r - 1
r = 1/(8 pi)
then h = 1/pi r^2 = 1/ pi[1/64 pi^2]
= 64 pi
pi r^2 h = 1
so r^2 = 1/(pi h) = etc, getting bored, check my arithmetic but you will find that the sphere has the minimum surface area per unit volume. That is why bubbles are round.
let volumes all be 1
sphere
v = (4/3) pi r^3 = 1
then r = [ 3/(4pi) ]^1/3
r = .620
a = 4 pi r^2 = 4.836
cube
v = s^3 = 1 so s = 1
a = 6 s^2 = 6 (so cools faster than sphere)
cylinder
LOL, depends on height to radius but
pi r^2 h = 1 so h = 1/(pi r^2)
and a = 4 pi r^2 + 2 pi r h
so
a = 4 pi r^2 + 2 pi r * 1/(pi r^2)
a = 4 pi r^2 + 1/r
look for max or min area for this volume
0 = 8 pi r - 1
r = 1/(8 pi)
then h = 1/pi r^2 = 1/ pi[1/64 pi^2]
= 64 pi
pi r^2 h = 1
so r^2 = 1/(pi h) = etc, getting bored, check my arithmetic but you will find that the sphere has the minimum surface area per unit volume. That is why bubbles are round.
1 answer