... either pure gold or it may have some silver in it .
archimedes figures out that the volume of the crown is 125 cm cubed and the mass is 1.8 kilograms.
he also knows that 1 kilogram of gold has a volume of about 50cm cubed and 1 kilogram of silver has a volume of about 100 cm cubed
is the crown pure gold why or why not?
Maths - the king asked archimedes if his crown is made from pure gold he knows that the crown is?
- Posted:
- 3+ months ago by cookiesan...
- Topics:
- gold, king, pure, figure, silver, crown, volume, math, maths
Details:
Answers (2)
1g = 1 gram = 1kg / 1000
1ccm = 1 cm^3 = 1 centimeter cubed
eq. = short for equation
u = gold, aUrum
r = silver, aRgentum
ρ = rho, density = mass / volume
According to the accompanying text of the assignment the density of
- gold is: ρ_u = 1000g / 50ccm = 20g / ccm
- silver is: ρ_r = 1000g / 100ccm = 10g / ccm
- the crown's alloy is: ρ_crown = 1800g / 125ccm = 14.4g / ccm
thereby telling us that
ρ_r < ρ_crown < ρ_u.
The crown, therefore, cannot be made of pure gold. So far, so easy.
But now it's getting interesting. The king's curiosity is excited and he inquires of Archimedes ...
Well, actually, I thought this purely qualitative answer to be rather unsatisfying. But as, unfortunately for us, Archimedes doesn't happen to be with us anymore, we are on our own. It's not that tough though, in fact, it's quite straight forward; so let's do this.
Question 1:
This will be related to the mass of the crown.
We need an eq. which describes, how many ccm of gold with 20g/ccm each plus how many ccm of silver with 10g/ccm each equals the mass of the crown.
u ccm * 20g/ccm + r ccm * 10g/ccm = 1800g ;; the ccm cancel just out
u * 20g + r * 10g = 1800g ;; simplify further: divide by 10g
Eq. 1) 2u + r = 180
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Question 2:
This will be related to the volume of the crown.
The second eq. has to describe how the volume of gold plus the volume of silver equals the volume of the crown.
u ccm + r ccm = 125ccm ;; divide by ccm
Eq. 2) u + r = 125
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(Each eq. alone has infinitely many solutions, but as we are going to solve them simultaneously, they constitute a system of mutually constrained parts.)
Eliminate r from the system of eq.s:
Eq. 3) = 1) - 2)
u = 55
Inserting this in eq. 2 yields
r = 70
And as we described the volume there, we know now:
55ccm of gold + 70ccm of silver = 125ccm, the volume of the crown.
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Inserting these values into the mass related eq.
u * 20g + r * 10g = 1800g yields
55 * 20g + 70 * 10g = 1800g
1100g of gold + 700g of silver = 1800g, the mass of the crown.