So you are starting with an ore of copper and trying to figure out the length of pure copper wire you can make.
The strategy I will use is to convert the mass of the ore into its equivalent mass of pure copper. Then, using the density, I will figure out the volume of pure copper that we have. A wire is essentially a cylinder with a round cross section but usually with a much greater lenght. Using the formula for the volume of a cylinder and the diameter given, we will figure out how long the wire is.
5.01 lbs of ore. The since the density of the copper is given in g/ml, I will eventually have to change the mass units to be the same. I find it easiest to work in grams, rather than pounds, so lets convert into grams.
2.2 lbs = 1000 grams
This means that 1000 grams / 2.2 lbs = 1
I can always multiply an equation by 1 and not change it, though it will seem to change because the units change.
5.01 * (1000 g / 2.2 lb) = 2277.27 g
Since only 66% of this is copper, if we purify it by smelting, we would have:
2277 g * 0.66 = 1503 g of pure copper.
(Remember that when multiplying by a percentage, you move the decimal 2 places.)
Density is the mass divided by the volume.
D = m / V
Our mass, m, is 1503 g and the density given is 8.95 g / ml.
Rearranging our equation, we get V = m / Density
V = 1503 g / (8.95 g/ml) = 167.93 ml
The volume, V, of a cylinder is given by:
V = pi * r^2 * h, where r is radius and h is height, or in our case, h is the length of the wire. (pi is approximately 3.1415.)
We don't have the radius, but we do have the diameter. The radius is half the diameter
r = diameter/2 = 6.304 x 10^-3 in /2 = 3.152 x 10^-3 in
I can see we will have an answer in either inches or cm. When I see ml, I think cm^3. I prefer to work in cm, so I will convert that radius to cm.
2.54 cm = 1 in
3.152 x 10^-3 in * 2.54 cm/ 1 in = 8.006 x 10^-3 cm
We are almost there.
From our cylinder equation, solving for h, the length of the wire, we get
h = V / (pi* r^2)
h = (167.93 cm ^ 3) / (3.1415 * (8.006 x 10^-3 cm)^2)
(I replaced ml with cm^3.)
h = 834,000 cm or 8,340 m.
The smallest number of significant figures used was 2 (66%), so I will round my answer to 8,300 m.