a-painting-attributed-to-vermeer-1632-1675-which-should-contain-no-more-than-962-of-its-original-c-3

Name: a painting attributed to vermeer 1632 1675 which should contain no more than 962 of its original c 3
Uploaded: 2020-04-10
Duration: 2 min 18 s
Channel: Numerade
Description: A painting attributed to Vermeer $(1632-1675)$ which should contain no more than $96.2 \%$ of its original carbon-14, contains $99.5 \%$ instead. About how old is the forgery?

Question

A painting attributed to Vermeer $(1632-1675)$ which should contain no more than $96.2 \%$ of its original carbon-14, contains $99.5 \%$ instead. About how old is the forgery?

Bobby Barnes · Accepted Answer

So we&#x27;re told some forgery has about 99.5% of its carbon 14 left. And what we want to do is determined. How old is thief Forgery. So they tell us. 99.5% left. Now, in example five. They tell us for C 14 dating that are constant, K, or our exponential decay equation should be the natural log to over 5730 reciprocal years. And at this point, we would just need to solve for tea. Once we plug, everything is so on the left side. We should have 0.995 Why not? Is equal to why not times e to the negative. And I&#x27;m going to just leave it as K for right now just so we don&#x27;t have to keep running that fraction out. So first thing we&#x27;re gonna do divide each side by. Why not for those canceled, Then take the natural log. So we&#x27;re going to get natural log of 0.995 and this is gonna be equal to negative K t Now we would divide by negative case to get T is equal to the natural log of zero point 995 over negative K, now plugging in K. That should give us 5000 or negative 5730 natural log of 0.995 all over natural log of two. And this would be in years. Now let&#x27;s actually approximate what This is so negative. 5000 730 times the natural look. 7300.995 is about 28.7. Divide that by natural log of two, and that gives about 41 point or years. So that is about how old are forged. Painting is.

Bobby Barnes · Answer

So in 1940 we were told that this cave painting was found and it had about 15% of the original carbon 14 remaining. We want to estimate how old is the painting? So let&#x27;s go ahead and see. So an example. Five. They tell us for carbon 14 dating that are K is going to be natural to over 5730 of reciprocal years and this fall&#x27;s exponential decay since it&#x27;s radioactive. So if we want to find how long it took for 15 years or 15% to be left, we would just plug in 0.15 Why not? Which is our initial amount on the left side, and then we&#x27;d want to solve or K. So I&#x27;m just gonna for tea. So I&#x27;m just gonna leave Kay, though as K for right now. And we can plug that into the variant. So first I&#x27;m going to do divide your side by. Why not get those cancel, take natural log and divide by negative k. So doing all that would give T is even to the negative metro log of 0.15 over. Okay, Now we can plug in what K is to get negative 5730 natural log of 0.15 over the natural log of two, and this would be in years. So let&#x27;s go ahead and see what this is approximately so negative. 5730 times natural like a 0.15 is about 10,000 or about 11,000. Then divide that by natural law goes to and that is going to give 15,682. So this is how old it was in 1940 so years mold in 1940. And then, if you want to be more specific, you can just go ahead and ad how many ever years it currently is to this to get its true H.

Bobby Barnes · Answer

So we want to determine how old this piece of charcoal is, given that it has about 44.5% of its carbon 14 remaining. Well, we know that carbon 14 dating deals with exponential decay. So first we&#x27;re going to have our equation. Why is even, too? Why not times e to the negative k t as the equation we&#x27;re going to use? And for carbon dating? K should be well, in example. Five. They actually go through this calculation to get that Kay is equal to. So this is coming from example. Five. I should say that it&#x27;s natural log of two over 5730 and our units with this, even though they don&#x27;t write, it, should be reciprocal years. Hurry now what we want to determine is hold. It is given that much less so. Let&#x27;s just go ahead and solve for tea to start after we plug that in. So it&#x27;s going to be zero point 445 Why not is equal to why not times e to the negative? Katie. So we would divide each side. Why? Why not take the natural log and we&#x27;ll get natural log of 0.44 By times, Negative word about times is equal to negative k t. So we would divide each side by negative K and we&#x27;re going to get T is equal to the natural log of zero point 445 over negative k. Those feral what K is first or bury it. Let&#x27;s just plug that in so we can get an exact answer, so it should be negative. Natural log of zero point or 45 over natural log of two times 5730. And this is going to be in years now since Kay was in reciprocal years. Now let&#x27;s multiply all this together. So the negative of the natural log of 0.445 is about 0.8. Divide that by natural log 02 That gives us about negative one about 1.1 and then multiply that by 5730. And this is going to give approximately 3393 years old or that piece of charcoal

Bobby Barnes · Answer

So in 1995 there was exponentially in in Peru that found what is called the ice maiden in, um, and they estimate the age to be 500 years. They want to determine about how much of carbon 14 was left. So now, in example, five they tell us for C 14 dating that are constant. K is natural aga to over 5730 reciprocal years, and this falls exponential decay since it&#x27;s radioactive. So if we want to find how much was left after 500 years, we would just do why of 500 and plug all of that. And so it&#x27;s going to be why not times e to the negative? It&#x27;s all right. This has 500 natural log of two over 5730 which should give us approximately. Let&#x27;s see so negative 500 times. Natural log of two is about negative 346. Divide that by 5730 was about negative 0.6. So why not times E to the negative 0.6 now exponentially ating? That should give about 0.941 AB zero point 941 Why not? Which that is going to imply. Then about 94 point A 4.1% left, and that was in 1995. Now they tell us if there is about 1% error in this process, what is actual maximum age? All right, well, they tell us, is about 500 years. So to find Max age, we would just multiply Bye one plus our error percent, which I&#x27;ll call just e. Actually, that&#x27;s probably bad number. Let&#x27;s use our sub e and we would just multiply by our H, so we&#x27;d end up with 1.1 times. Well, they said it was about 500 years, so we would expect adding Max an age of 505 years. And so this was in 1995 when they actually found it.

Bobby Barnes · Answer

So they tell us to use carbon 14 dating to help us determine when this animal died from the bone that they found. So they said they found the bone in the year 2000. And the three cases that we&#x27;re going to have is when they&#x27;re 17% 18 and 16% of our carbon 14 left now in exercise five, they actually give us that four C 14 k is going to be natural Aug two over 5730 reciprocal years and we know are equation or radioactive isotopes should follow. Why not times e to the negative? Katie. So since the only thing that&#x27;s going to be changing in these three equations is why of tea, I&#x27;m going to first write this as there&#x27;s gonna be p times. Why not? Which P is going to be our percent? This is going to be equal to Why not times e to the negative. Katie and I know a comm plugin K, but I&#x27;m just gonna wait until the end to do that just so we&#x27;re not having to work with that fraction Now. Let&#x27;s solve for tea in this, and then we could just plug in our three percentages as decimals. So for a should be 0.17 b 0.18 and and C 0.16 Yeah, So let&#x27;s go ahead and solve for K, though I mean for teeth. So we first divide by. Why not? That cancels. Then we&#x27;d want to take the natural log on each side, which is going to give us natural log of P is equal to negative. Katie. Then we could divide by negative K on each side. So we&#x27;re going to get t is equal to actually let me write this over here. So this implies that t easy to the natural log of p over negative K and negative K would be equal to negative natural log of P and let me expand the screen a little bit So natural. Aga P Times 5730 Over natural log of two And this should be in years. Now let&#x27;s go ahead and plug in P is equal to each of those that should have been 0.17 not 0.0. But let&#x27;s plug those three values and now so for a T is going to be equal to So it&#x27;s the natural log of Well, let&#x27;s actually just plug it in in first of their appointed 17 and then 570 over natural law. Go to and I don&#x27;t know why. I want to keep writing that extra zero right there, but this should give us approximately. Let&#x27;s see so negative. Natural log of 0.7 is about 1.7, multiplied by 570 divide by natural log of two. So this is going to give what about 14,600 and 48 years? But technically, this is just telling us for tea. We said we were found in the year 2000 so let&#x27;s go ahead and actually do 2000 minus this to get the actual year that this died so 2000 minus. This should be something around. So if we subtract, is about so would be negative. 12,648. But for negative years, we would just call that BC still too big. So that&#x27;s going to be the first one. Now we can do the same for the others. Now we&#x27;ll get be T should be negative natural log of 0.18 times 5730 over the natural law of two. So it&#x27;ll be the same thing that we did before. But now just 0.8, and after we divide everything, this should give about 14,000 176 years. So now if we do 2000 minus 14,000 176 this would give negative 12 176 But when we have negative years is going to just be, be see and then lastly for 16%. So it&#x27;s negative natural look of 0.16 5730 over the natural log of two, and this should give about negative Natural Lago 0.6 is about 1.8 times 15 5730 divided by natural log of two. So that&#x27;s going to give 15,140 Niner approximately 149 years ago. Or that&#x27;s how long it took. So again we would do minus 2000 and that&#x27;s going to give 13,140 nine b. C. So that would be the year that the animal died in that case,

Bobby Barnes · Answer

So we&#x27;re told that this frozen human mummy was estimated to have died about 5000 years ago. And we want to determine how much of the original carbon 14 remained at the time of his discovery. Well, in example five, they tell us that for C 14 carbon dating, we will have our kay is equal to natural log of two over 5730. And the mount left will follow just the typical exponential decay equation. So for 5000 years ago, that means we&#x27;d want to plug 5000 in to that, so we&#x27;d have why 5000 is equal to. So why not times E to the negative K. So I was going to be natural log of two over 5000 730 and then we multiply this by 5000. So let&#x27;s see what that some natural law goes to times 5000 is about 3465. Then divide by negative 5000 730. So this is going to be why not times e to the negative zero 0.6 about and then exponentially eating. That should give about zero point by for six times. Why not? And so that we multiply it by 100 should be our percent left, which would give us about 54.6 per cent of those C 14 that would be left.

Problem

The answers to most of the following exercises ar…

Problem 48 Easy Difficulty

A painting attributed to Vermeer $(1632-1675)$ which should contain no more than $96.2 \%$ of its original carbon-14, contains $99.5 \%$ instead. About how old is the forgery?

Answer

About 41.4 years.

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Joel Hass, Christopher Heil, Przemyslaw Bogacki

University Calculus: Early Transcendentals

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About 41.4 years.

University Calculus: Early Transcendentals

Grace H.

Catherine R.

Anna Marie V.

Heather Z.

Integration Review - Intro

Definite vs Indefinite

Joel Hass, Christopher Heil, Przemyslaw BogackiUniversity Calculus: Early Transcendentals

Grace H.

Catherine R.

Anna Marie V.

Heather Z.

Joel Hass, Christopher Heil, Przemyslaw Bogacki

University Calculus: Early Transcendentals