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Problem 32 Hard Difficulty

An isotope of sodium, $ ^{24} Na $, has a half-life of 15 hours. A sample of this isotope has mass 2 g.

(a) Find the amount remaining after 60 hours.
(b) Find the amount remaining after $ t $ hours.
(c) Estimate the amount remaining after 4 days.
(d) Use a graph to estimate the time required for the mass to be reduced to 0.01 g.

Answer

a) 0.125 $\mathrm{g}$
b) $y=2\left(\frac{1}{2}\right)^{t / 15}$
c) 0.0237 $\mathrm{g}$
d) 114.6 $\mathrm{h}$

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Video Transcript

all right, We're doing 1/2 life problem and we know the half life of the substance is 15 hours, and we know we start with two grams. And so when 15 hours elapses, there's now one gram, half of what was started with another 15 hours goes by and there's half of it. Another 15 hours, another half. And then finally, when we get to 60 hours, which is what we're interested in for Ah, this problem, we're down to 1/8 of a gram. So the answer to part A is when 60 hours have elapsed, there's 1/8 of a gram left. Okay, the next thing we want to do is figure out kind of a generalization so that we can know how many grams a remaining after t hours. So looking at the pattern we've seen so far, notice that we have started with two. And then we multiplied it by 1/2. We multiplied it by 1/2 again 1/2 again on 1/2 again. So the ex pony on 1/2 keeps increasing. And how does that exponents relate to the number of hours that have elapsed? So if you take the number of hours and divide it by the half. Life 60 divided by 15 You get four the number of hours divided by the half. Life 45 divided by 15 You get three etcetera so we can generalize this. And so if t hours have elapsed then we have the number of hours t divided by the half, life as our exponents. So this gives us our formula for the number of grams remaining after T hours. Now we can use that formula to figure out how many grams would be remaining after four days have gone by. So four days have gone by. 96 hours have gone by, so we can use 96 our formula and put that into the calculator and we get 0.237 grams. And lastly, we want to know how long it would take for the mass to be reduced 2.1 grams, 1 1/100 of a gram. So we grab a calculator and we type are function in two times 1/2 to the X over 15 power and then we also type y equals 0.1 So that would be a horizontal line at a height of 0.1 and that represents that level of 1 1/100 of a gram remaining for my window. I fiddled with the numbers for a while until I came up with a good view of everything. I chose to go from negative 1 to 150 on my X axis and negative 1500.1 2.5 on my Y axis. And those numbers convey Ari. You just want to be able to see both your exponential decay graph as well as your horizontal line, and you want to be able to see the intersection point. So now let's find that intersection point so we can go into the calculate menu and choose number five Intersect. We have the cursor on the first curve, so we press enter. We have the cursor on the second curve, so we press enter and then we go over toward the intersection point and we press enter. Now, looking at the bottom of the screen, we can see that X equals 1 14.66 when y equals 0.1 So that tells us that it's going to take approximately 114.66 hours to get to this level