Cylinders/Spheres

Question

In manufacturing buckshot (lead cylinders) of radius 1inch and height 8inch are melted and recast a spherical shot of diameter 1/16inch. How many shot can be made from 1 cylinder, assuming no waste?

Answer 1

Hi Jess!

This is a good question. What we really want to know is the maximum volume of in spheres that can be made from the volume of one cylinder.

Volume of Cylinder = πr²h = π(1)²(8) = 8π in³

We also need to know the volume of each sphere to determine how many can be made:

Volume of Sphere = (4/3)πr³ = (4/3)π(1/32)³ = π / 24576 in³

To find the amount of spheres that fit into the cylinder we just divide the volume of the cylinder by the volume of each sphere and drop the remainder since you can't have part of a sphere as a product.

Volume of Cylinder / Volume of Sphere = 8π / (π / 24567) = 8 (24567) = 196,608 spheres.

Hope this helps!

-Nathan

Answer 2

Lets start with whats given.

We start with cylinders of radius=1, and height=8.

We turn this cylinder into multiple spheres of diameter=1/16.

It says we melt a cylinder, so how much material would we get from it? Lets find the volume.

Volume of a cylinder is π*radius²*height. Here this is π*1*8, or 8π.

So we have 8π material to work with per cylinder. The question says we only melt one cylinder, so this is all we have.

Lets find how much of this material we need to make one of these 'lead shots' (a sphere).

The volume of a sphere is (4/3)*π*radius³. We have a diameter of 1/16, and radius is half of diameter, so we do 1/2 * 1/16, and get radius is 1/32. We then plug this into the formula and have (4/3)*π*(1/32)³.

The number we get from that is pretty ugly, but we have calculators, so lets let it sit for a minute.

Now, the question asks how many shots can be made, so this is a division problem. We divide the amount we have, by the amount we need to make each sphere, and we will get number of spheres:

(Material We Have) / (Material needed for a ball) = Number of Balls.

We do this, and get 8π/((4/3)*π*(1/32)³).

We need a calculator for this--and when we plug it in, we get 196,608.

This means that with one 'buckshot' (cylinder), we can make 196,608 'lead shots' (sphere).

Hope this helps!