The discussion revolves around calculating the area that gold leaf from a gold bar can cover, given its thickness and the volume of the bar. Participants explore the conversion of units and the application of formulas related to volume and area, with a focus on mathematical reasoning and unit conversions.
Participants generally agree on the need for unit conversions and the application of the volume-area relationship, but there is some uncertainty in the calculations and final results, particularly regarding the conversion process and the interpretation of the final area covered.
Some participants express difficulty with the mathematical aspects, and there are unresolved steps in the calculations, particularly in ensuring accuracy during conversions and the handling of large numbers.
Your first task should be to express everything in the same units. So how many nm are there in a dm?Math_312 46 said:The gold pavilion in the city of Kyoto is covered with gold leaf. Gold leaf has a thickness of 100 nm (nanometres) and is made from gold bars. A gold bar has a volume of 0.65 dm^3. How many square metres can the gold leaf from a gold bar cover?
Is the answer 0,01m^2?Opalg said:Your first task should be to express everything in the same units. So how many nm are there in a dm?
Next step: how many nm^3 in a dm^3? (You'll be getting some large powers of 10 at this stage.)
If the volume of a gold bar is X nm^3 then the formula V=Ah (with H = 100nm) tells you that the area covered by the gold leaf will be given by 100A = X.
Finally, that area A is in units of nm^2. You need to convert that to m^2 to get the result in square metres.
That really doesn't seem very likely, does it? In fact, 0.01m^2 is one hundredth of a square metre. That is less than the size of a sheet of paper. If you take a gold bar and hammer it out until it is microscopically thin (which is what gold leaf is), you would expect it to cover quite a large area, certainly many hundreds of square metres.Math_312 46 said:Is the answer 0,01m^2?
A decimetre (dm) is one tenth of a metre. A nanometre (nm) is one billionth of a metre. So there are $10^9$ nm in a metre, and therefore $10^8$ nm in a dm. When you cube that in order to get a measure of volume, you find that there are $10^{24}$ nm^3 in a dm^3. So a volume of $0.65$ dm^3 is equal to $0.65\times10^{24}$ (or equivalently $6.5\times10^{23}$) nm^3.Math_312 46 said:If this is wrong can you help me?
Opalg said:That really doesn't seem very likely, does it? In fact, 0.01m^2 is one hundredth of a square metre. That is less than the size of a sheet of paper. If you take a gold bar and hammer it out until it is microscopically thin (which is what gold leaf is), you would expect it to cover quite a large area, certainly many hundreds of square metres.
A decimetre (dm) is one tenth of a metre. A nanometre (nm) is one billionth of a metre. So there are $10^9$ nm in a metre, and therefore $10^8$ nm in a dm. When you cube that in order to get a measure of volume, you find that there are $10^{24}$ nm^3 in a dm^3. So a volume of $0.65$ dm^3 is equal to $0.65\times10^{24}$ (or equivalently $6.5\times10^{23}$) nm^3.
Can you continue from there?
You left out the decimal point, so it should be 6.5 e 21 nm^2. But the question asks for the answer in square metres, so you still have to divide by $\bigl(10^9\bigr)^2 = 10^{18}$ to get the answer in those units.Math_312 46 said:Sorry in advance but I find maths very difficult. But according to my calculations based on your explanation, it came out like this:
6.5 x 10^23 / 100 = 65 e 21 (65 and 21 zeros)
And thus I get the answer that 65 e 21nm² can the gold leaf of a gold bar cover?
Opalg said:You left out the decimal point, so it should be 6.5 e 21 nm^2. But the question asks for the answer in square metres, so you still have to divide by $\bigl(10^9\bigr)^2 = 10^{18}$ to get the answer in those units.
Nearly there! You have one too many zeros. It should be 6.5 e 3 = 6500 m^2.Math_312 46 said:Ah, I see. so if I convert 6.5 e 21 nm² to m², which then becomes 65000 m². Is this the answer or is something wrong?
Opalg said:Nearly there! You have one too many zeroes. It should be 6.5 e 3 = 6500 m^2.
Yes. :)Math_312 46 said:So to be sure the answer is 6500 m^2 of the whole question?
Opalg said:Yes. :)