Assuming that there are no empty spaces between the gumballs, how many gumballs are needed to completely fill the glass jar?

Question

Clyde wants to fill a glass jar with gumballs. The glass jar is the shape of a cylinder with height 10cm and radius 6cm. The gumballs are spheres of identical size Each gumball has a surface area of 12 cm^2.

Assuming that there are no empty spaces between the gumballs, how many gumballs are needed to completely fill the glass jar?

Round each calculation to 2 decimal places.

Answer 1

This is not a very realistic problem because we are told to "no empty spaces between the gumballs" meaning these would need to be "fluidic" gumballs I suppose.

Also, it is not a good practice to round each calculation and then to use that rounded number in your other calculations - it adds round-off error in each subsequent calculation. But I'll follow the instructions anyway and do so:

The surface area of a sphere is 4πr² so:

12 = 4πr²

3/π = r²

0.95493 = r²

r = 0.9772 cm = 0.98 cm

The volume of a sphere is 4/3πr³ so the volume of one gumball is 4/3π(0.98)³ = 3.9425 cm³ = 3.94 cm³

The volume of the gπlass jar (cylinder) is πr²h = π(6)²•10 = 1130.973355 cm³ 1130.97 cm³

The number of gumballs = (volume cylinder)/(volume gumball) = 1130.97/3.94 = 287.05 gumballs

Answer 2

This is kind of an odd question, in terms of how one is supposed to interpret it and the rounding directions (ie round to two decimals at each step, or round the final answer); here's how I did:

Since we are told that there are no empty spaces between the gumballs, the entire volume of the jar will be filled.

The volume of the jar is given by

pi (r^2)(h) = pi(6^2)(10) ≈ 1130.97 cm^2.

We now want to find the volume of a gumball. We are told that they have a surface area of 12 cm^2, and so, using the surface area formula for a sphere (SA = 4*pi*r^2), we know that

12 = 4*pi*(r^2), hence r^2 = 3/pi, so r=sqrt( 3/pi) ≈ 0.98 cm.

Then, using the volume formula for a sphere,

V = (4/3)*pi*(r^3), we have the volume of a gumball is about (4/3)*pi*(0.98)^3 ≈ 3.94 cm^3.

Dividing the volume of the jar by the volume of a gumball gives us

1130.97/3.94 ≈ 287.05

Hence, the jar will be filled by about 287 gumballs (not sure if the rounding directions apply here, since you can't have 0.05 gumballs).

As an aside, believe it or not this problem is studied by people, allowing for spaces between the gumballs (since there would be spaces in real life), they're called "sphere packings"

Answer 3

A = 4πr²

12 = 4πr²

3 = πr²

0.9772 ≈ r

28 gumballs make up the first layer.

10 / 1.95 = 5.11

(28)(5) = ?