Copper and lead in a box.
A box contains a mixture of small copper spheres and small lead spheres. The total volume of both metals is measured by the displacement of water to be 427 cm3 and the total mass is 4.36 kg. What percentage of the spheres are copper? Remember significant figures!
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This is a problem of proportions and average density. You can find the density of both metals online:
Density of Copper: 8.94 g/cm^3
Density of Lead: 11.34 g/cm^3
The easiest way to do this is to say x is the proportion of the copper spheres out of the total number of spheres, making the fraction of lead spheres equal to 1-x (because we only have lead and copper spheres).
This would make the average density the following (in g/cm^3):
8.94x + 11.34(1 - x)
To get total mass, we simply multiply the volume the spheres take up by their average density:
m = dv
From the problem and our expression for average density, we get the following equation (keep in mind mass needs to be in grams):
4360 = (8.94x + 11.34(1 - x))*427
Simplifying the right side:
4360 = 427(8.94x + 11.34 - 11.34x)
4360 = 427(-2.4x + 11.34)
4360 = -1024.8x + 4842.18
Now, we can solve for x, first by subtracting, then by dividing:
-482.18 = -1024.8x
0.471 = x
Notice our calculated fraction of copper spheres has 3 significant figures, like every number given in the problem.
Because they ask for a percent of copper spheres, you just multiply x by 100% and get the following:
Percent of copper spheres = 47.1%
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