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The radioactive decay of Unobtainium can be measured by flt) = A e^(-t/8,824). wherc A is the initial amount and t is in vears_ How long does it take 1g of Unobtain...

Question

The radioactive decay of Unobtainium can be measured by flt) = A e^(-t/8,824). wherc A is the initial amount and t is in vears_ How long does it take 1g of Unobtainium to decay to 0.00001g?

The radioactive decay of Unobtainium can be measured by flt) = A e^(-t/8,824). wherc A is the initial amount and t is in vears_ How long does it take 1g of Unobtainium to decay to 0.00001g?



Answers

Radioactive Decay Suppose the quantity (in grams) of a radioactive substance present at time $t$ is

$Q(t)=1000\left(5^{-0.3 \tau}\right)$

where $t$ is measured in months.
(a) How much will be present in 6 months?
(b) How long will it take to reduce the substance to 8 g?

We're doing a DK problem. So we're using the model M equals M, not each of the K T, which is the same as the population growth model. We just are using em instead of P. And the value of K is going to be negative instead of positive since it's not growth. Okay, so what we want to do is find the half life and the given information is that it takes one year to decay to 94.5% of its original amount. So we're going to use that piece of information to first find a model and then once we have the model, we confined the half life. So we're going to substitute 0.945 of the initial amount 0.945 times and not and for the final amount, will substitute am knockin for em, not keep it the same on will substitute one year and for the time now we divide both sides by I am not and we have 0.945 equals each of the K and we take the natural log of both sides and we get the value of K so we can substitute that into our model and we have em of tea equals m not times e to the natural log of 0.945 times time. So we'll use that model for the rest of the problem. Now we're gonna use it to find the half life and half Life is the time it takes to decay to half the amount. So if we're starting with them, not we would be ending with 1/2 am not when half am not equals. M not times e to the natural log of 0.945 t divide both sides by I am not, and we're left with 1/2 and I'm just gonna write it as 0.5 and then we take the natural log on both sides and then to get t by itself, we're going to divide by the natural log of 0.945 and we put this in the calculator on. We get approximately 12.25 years. So the half life of this substance is about 12.25 years. Now we want to do part B, and it's very similar to party. But instead of having half of it left, we have 20% of it left. So 20% of the original mount would be 200.2 times m. Not so we'll substitute that into our equation for em. 0.2 am not is equal to am not times e to the natural log of 0.945 times time and we're solving for time again because we want to know how long will it take? Divide both sides by I am not. And we have your point too. And then we take the natural log of both sides and then we divide both sides by the natural log of 0.945 to get t. Then we put it in the calculator and we get approximately 28.4 point 45 years. That's the time it takes to decay to 20% of what it started with

By using the given function, FT. We will try to get ah, an exponential function for the amount off plutonium. After 30 years and ah, half old house life off plutonium is given. It is 24 360 years. And after this health life off the elements half Oh, it's ah left eso uh, initial amount off the plutonium is given here £10. So, um, after half life ah, it's half will be left that will be £5. So, uh, let's use them. And let's find Constant Keene s. Oh, let me write Five equals 10 times e to the k times t is its half life, and that is 24,316. And then here had been Dwight both side by 10. And let me simplify them by five. We will get when in to so 1/2 equals Here it's into five. It will be e to the 24 316th and, uh, well over two means is opened. Five So open in five equal into the 24 36 them on. Let's take on both sides A natural law. Garrett him, Andi. Then we will get here, Ellen. 0.5 equal power goals here. 24 360 times. Ellen E. And you know, Ellen is one, so oh, to hear report Got t k. Sorry. And here we have K and K find K. That's right. Both sides half life. So do I. Did it and then on. And then everyone got approximate value off K by using calculator. When you can quit this fraction, it will be negative. 4.0, 285 And then, uh, the general, um, exponential function off. Uh, plutonium believe f t equals in issue. Amount is 10 times e to the power off. Katie, that is a minus a 0.0 or 285 t and, uh, here, uh, after how many years that amount off plutonium will be £2 to find it here. That's right. Ftv as to so equals 10 times e to the power off miners. 0.285 team. And then hear that divide both sides by 10. So when we divide, we will get here by tour. Let me simplified. Will be won over five 1/5. Equal. Simplify tense e to the minors all point to a five T and the tape both sides Nature, luxury Tim get team. So Ellen 1/5 equal miners O J. Times, T Times, Ellen E. And Ellen. He is one. So fine team. That's so I bought side by k So Thiel's and calculate there's with calculator. So the Americans to five. The approximate value off T will be it is six or 70 to 2472 years after the amounts off battalion will be £2.

By using a function f t. We will figure out an exponential function for the amount of platinum after two years and the initial, um, amount off platinum is given. It is 10 and after half life it's half left so it's half will be five so five equals 10 times e to the que times TT chose time here. So it's halftime you will use It is 24 360 years. So let's do I go outside by 11th full and here runny simplify We will get 1/2. 1/2 equals 10 tin can self. It will be he to the Power Grove when the war is to be under 60 k And let's take both sides natural Worried him and we won't get Helen. Well, the word to the girls are goes here Helen eating. And you know that Ellen, he is one. So, um, you will get here where we divide both sides by 2040 six Tim 14 to 6 temp. When you calculate is with calculator, the approximate value off K will be here. We can simplify. It is well, negative or point. Oh ho, 285 and, um, our function will be in equals here. Ft for the amount of plutonium after two years, uh, was initial amount is £10 10 times e to them. Okay. You know, the Syrian foreign kids for point. 00 toe 85 t and ah, question. Also asked that when will be the amount off pretending will be £2. So instead, off ft, you will write to and you will be able find when this will be happen. 285 Team dried roadside by 10 and simplify. So 1/5 equal you insult. He took the minus. O toole 85 teen. Then take both sides. Nature a little very Tim. So you will get a man 1/5 equals what comes here? Five team times. Ellen Dean. Felonies vom. So do I. Both side by K. And here dried it. So calculate is with calculator and you will get their approximate value off team if they're six or so. I wanted to after 6000 472 years, the amount off the telly, um, about £2

Okay, So in this question, they're saying rodeo 105 is made by new Trump Environment of Brittany and 104 And they want us to write the balance nuclear equation for the formation of rodeo 105 So we have routine IAM 104 They say it was Newsome bombardment. Brian informs rhodium one of five. So the first thing that we should dio is look at the period table and determine what the attack number of rhodium and Ruth idiom is. So we see the routine IAM is 44 and we see that rhodium is 45. Now, in order to find the missing particle, we have to balance out the number two topic number and the mass on both sides. So 44 from the are you cluster zero from neutrons. It's equal to D 45 from the ah rodeo, and I wanna bounce that. We need to have a minus one on the right side. So you put a minus one here. Now we know that at minus one is a beta particle. So let's look at the master and we have 104 from the RU plus t one from the of neutron Crimea 105 from the our age. So we have the ad zero. So beta particle has a massive zero and that's expected now 42nd part. They're asking us to calculate how long it will take for 99% of the Rh to decay. So they tell us the half life is 4.4 years. So T half life is 4.4 years. They're saying 99% decay, right? So how much is left? So 1% is left. So the equation we need to use is to the end power and is the number of half life that's passed is equal to D amount left in decimals. So since we have 2% we just divided by 100 we get 1000.1 So 0.1 Now in order. Nicely end. We can take the log of both sides, right? One of the rules is that if you want something, give appendix exponents through the front. Now, how do we isolate ad? Well, we divide both sides by a log of one half, right? The log one have crosses out. You get em is equal to 6.644 Now, this is the number of half life that needs to pass in order for 1% of it to be left. So how long is that? Well, six points. Ah, 644 times it by the 4.4 years. And we get 29 points to three years.


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