Solving Archimedes' Problem: Is the Crown Gold?

[tandoorichicken]

A king's crown is said to be solid gold but may be made of lead and covered with gold. When it is weighed in air, the scale reads .475 kg. When it is submerged in water, the scale reads .437 kg. (a) is it solid gold? (b) if not, what percentage by mass is gold?

density of gold = 19.3 * 10^3 kg/m^3
density of lead = 11.4 * 10^3 kg/m^3

 

[ShawnD]

Originally posted by tandoorichicken
A king's crown is said to be solid gold but may be made of lead and covered with gold. When it is weighed in air, the scale reads .475 kg. When it is submerged in water, the scale reads .437 kg. (a) is it solid gold? (b) if not, what percentage by mass is gold?

density of gold = 19.3 * 10^3 kg/m^3
density of lead = 11.4 * 10^3 kg/m^3

I'll do units in g/cm

mass in water:
m = pv (crown) - pv (water)
0.437 = pv - 1v
0.437 = (p - 1)v
v = 0.437/(p - 1)

mass in air:
m = pv
0.475 = p(0.437/(p - 1))
0.475 = 0.437p/(p - 1)
0.475p - 0.475 = 0.437p
0.038p = 0.475
p = 12.5

So it's not gold.

density composition, x is the amount of gold
p = 19.3x + 11.4(1 - x)
12.5 = 19.3x + 11.4 - 11.4x
1.1 = 7.9x
x = 0.1392

The crown is roughly 14% gold and 86% lead.

 

[HallsofIvy]

ShawnD- Don't go doing all of TandooriChicken's homework for him! Make him show that he has at least attempted the problem himself!

 

[ShawnD]

Ok I won't do that next time.
Think of it this way though, if tandoori went through the trouble of posting it here, he is probably serious about wanting to know the answer. Most teachers don't even mark you based on your homework; in all of my high school math, chem, physics and biology classes, homework was never checked or marked.

IIRC, tandoori said he was homeschooled. His instructor is (most likely) not university educated or even familiar with doing slightly difficult math procedures such as making that equation to find the percentages of each element.

 

[tandoorichicken]

homeschooled? don't remember ever saying that...
hmmm maybe you were talking wit a different tandoori shawnd.
anyways, thanks for the help.
and yeah, most of the problems i post here i either don't know how to start or got lost halfway through. I am no slacker hehe

 

[tandoorichicken]

shawnd, are you sure that the second part is mass percent? because your post says density comp.

 

[ShawnD]

I misread the question, I'm sure you can figure that out with all the info given and solved for.