You are relaxing at home, working on your coin collection....

You are relaxing at home, working on your coin collection. You have learned that collectible coins are sometimes plated with gold to enhance their beauty and value. On an online site that sells collectible coins, you see a commemorative quarter-dollar advertised for sale at $4.98. The information on the site says that the coin has a diameter of 24.6 mm and a thickness of 2.10 mm, and it is completely covered with a layer of pure gold 0.175 µm thick. You know that gold is valuable, so you come up with a scheme by which you would separate the gold from the base coin and sell the gold at the current market price. You begin your scheme by buying the advertised coin at $4.98. (Assume the density of gold is 19.3 g/cm³.) (a) What is the value of the gold for a single such coin (in cents)? Assume that the price of gold is $40.00 per gram. Carefully evaluate the volume of gold on the faces and edge of the coin. (b) What If? What would be the cost (in dollars) of the gold to produce the coin if the coin were made entirely of gold? Check your equation for the area of the coin; to what power is the radius raised?

Transcript

00:01 So in this problem, we're evaluating a gold coin.

00:04 We know that the diameter of the gold coin is 24 .6 millimeters, and that is a symbol for diameter, and that its height is 2 .1 millimeters.

00:26 So in part a, we see that this is plated in gold, which means every surface is going to be covered in this thin layer of gold, which we know is 0 .17.

00:38 And five micrometers thick.

00:47 So first we figure out what kind of variable we need to do.

00:51 And we know that the cost of gold is $40 per gram.

00:57 So that means we need some kind of weight because by dimensional analysis we want dollars.

01:02 So if we want to just the dollars, if we multiply by some weight, the grams cancel, and then we'll be left with dollars.

01:09 So we want weight, but we don't know that yet because we don't know how much gold.

01:12 We're going to need to find the volume of gold to do that.

01:16 So instead of just weight, we're going to need our density as we were given, and then find some volume.

01:23 And this is where all of our figuring out is going to go.

01:27 So when you look at what kind of area that we have to cover here.

01:31 Firstly, we have the top and bottom of the coin.

01:34 So these are both circles.

01:36 They're going to say they have some radius and an area of a circle, pi r squared on each of them.

01:44 There's also then the thickness, and that goes all the way around the circumference of the coin.

01:52 And so that area, all the way around the coin, is simply the circumference times the thickness.

02:03 So our circumference is pi times the diameter that we know, and then the thickness was given as that is 0, as that 2 .1 millimeters.

02:17 So that is the basic way.

02:19 If you think carefully about this, though, you'd realize that we're actually kind of missing a bit right around the corners.

02:26 So like, let's zoom in to the edge of the coin and say that there is some thickness of gold on it.

02:35 What we've measured here is the area on the top face here and then the area around the edge.

02:42 We're also going to be multiplied by the thickness of the gold, but we will get there in a second.

02:47 But notice that both of these things we've done haven't accounted for this square, this little edge in the middle, or this edge.

02:55 Now this is a very, very tiny bit because we see that the thickness of the gold is very small compared to the other dimensions.

03:04 Micromedars are 10 to the native 6th, 1 millionth of a meter compared to millimeters or centimeters.

03:10 And so this corner is very small and we're going to neglect it.

03:13 This is a fairly standard thing if you're dealing with thin coatings like paint.

03:17 If you had something more significant, like let's say something like this, you would have to consider the edge because this corner is much bigger.

03:27 And you see if we'd neglect it, we would lose like half of the volume.

03:35 But basically, so we have our, we want to find our cost.

03:40 And we know that that's going to be equal to our dollars per gram.

03:45 And then the weight, so the density, which was given is, let's put the units here because the units are actually really important to figure this out to make sure that we are keeping everything in track.

03:56 So our cost is equal to $40.

04:00 Per gram, and then we know that gold is 19 .3 gram per centimeter cubed.

04:13 Then we see the grams cancel, so we're left with the centimeters.

04:19 That way we need to figure out.

04:20 So now we want to find the volume.

04:22 So the first thing we're going to do is multiply by the thickness, the 0 .175 micrometers.

04:31 Now this is actually, we know that a micrometer is 10 to negative 6 meters.

04:43 And if we multiply that by 100 centimeters in a meter, we would end up with 10 to the negative 4.

04:52 We're going to leave everything in centimeters because we have cost in centimeters, and i'm just going to leave it in that.

04:58 You could also go to base units to meters, but that would be an extra conversion.

05:02 And so here we'll just leave it in centimeters.

05:06 So the 0 .175, we figured out they need to be 10 to the 4 or times 10 to the negative 4 centimeters.

05:17 So we've canceled one of the centimeters, but we can't cancel the rest yet.

05:20 So now what we need to do is we have the thickness.

05:22 We need to consider the area that is covered...

Question

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