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.
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.
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.
To determine which shape takes the longest time to cool down, we need to understand the concept of surface area and its relation to cooling.
When an object cools down, it loses heat through its surface. The rate at which an object cools down is proportional to the surface area through which heat can dissipate. Therefore, the shape with the largest surface area will typically cool down the fastest.
Let's compare the surface areas of a cube, a sphere, and a cylinder:
1. Cube: A cube has six identical square faces. The total surface area of a cube can be calculated by multiplying the length of one side by itself and then multiplying the result by six: Surface Area = 6 * (side length)^2.
2. Sphere: A sphere has a curved surface all around. The surface area of a sphere can be calculated by multiplying the radius squared by 4π: Surface Area = 4π * (radius)^2.
3. Cylinder: A cylinder has two flat circular faces and one curved surface around the sides. The surface area of a cylinder can be calculated by adding the areas of the two circular bases and the area of the curved surface: Surface Area = 2π * radius^2 + 2π * radius * height.
Based on the formulas for surface area, we can see that the sphere has the smallest surface area for a given volume. The cube will have the largest surface area, and the cylinder will have an intermediate surface area.
Therefore, the sphere will take the longest time to cool down since it has the smallest surface area and thus less area for heat to dissipate. The cube, with the largest surface area, will cool down the fastest, and the cylinder will cool down somewhere between the sphere and the cube.