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caesium-137 is a radioactive tracer element used to study upslope soil erosion and downstream sedimentation it has a half-life of approximately 30 days so this half-life of 30 days this means that if you were to start with let's say if you were to start with 2 kilograms of cesium 137 that 30 days later 30 days later you're going to have 1 kilogram of cesium 137 that the other kilogram has decayed into other things assume and if you worried another 30 days you would have half a kilogram of cesium 137 assume that the amount a in becquerels of cesium 137 in a soil sample is given by the exponential function a is equal to C times R to the T where T is the number of days since the release of cesium 137 into the soil and C and R are unknown constants so this bears some explaining what is this Becquerel business so normally if I were to talk about the amount of some element I'd probably be thinking in terms of mass and I might talk in terms of kilograms but some people might also be referring to the amount of this radioactive substance in terms of the amount of radioactivity it produces and becquerels is the international unit of radioactivity named after Henri Becquerel who co-discovered radioactivity with Marie Curie so you could consider this the amount of the amount of cesium 137 that causes a becquerels of active radioactivity but either way we can just think of it as a as a quantity but it's really the quantity that causes a background of activity so just to be clear the amount is given by the exponential function a is equal to let me just rewrite it a is equal to C times R to the T power where T is the number of days since the release of the cesium 137 in the soil and C and our unknown constants fair enough so let's just be clear this is days days since release in addition assume that we know that the initial amount of C's 1:37 released in the soil is eight becquerels is eight becquerels solve for the unknown constants C and R so the initial of the soil that's when T is equal to zero when no days have passed so we could say that the amount at time zero well that's going to be equal to C times R to the 0 power which is just going to be equal to C times 1 which is equal to C and they tell us what a of 0 is they say a of 0 is 8 is 8 becquerels so this is going to be equal to 8 so our constant here the C is just going to be equal to 8 what is the value of the constant we can just write 8 right over there then they ask us what is the value so the value of the constant C is 8 what is the value of the constant our round to the nearest thousandth so we're starting with 8 so a of 0 is 8 how much are we going to have after 30 days and the reason why I'm picking 30 days is that is the half-life of caesium-137 so a of 30 remember our T is in let me just switch colors just for fun remember T is in days so a of 30 so after 30 days I'm going to this is if I want to use this this this the formula right over here if I wanted to use the description of this exponential function we already know that C is 8 it's going to be 8 times R to the 30th power which is going to be equal to what well if we started with eight 30 days later we're going to have half as much we're going to have 4 becquerels we're going to have 4 and now we can use this to solve for R so you have 8 times R to the 30th power is equal to 4 divide both sides by 8 you get R to the 30th power is equal to 4 over 8 which is the same thing as 1/2 and then we can take the 130th we can raise both sides to the 130th power R to the 30th but then the 1 then you could think of it the 30th root of that or raising that to the one 30th power that's just going to give us R is equal to 1/2 to the one thirtieth power and that is something that's very hard to compute in your head so I suggest you use a calculator for that and they they hint because we're going to round to the nearest we're going to round to the nearest thousandth so let's get a calculator right out and so we're talking about 1/2 to the 1 over 30 power 1 over 30 power so we get zero point nine seven seven one five nine nine keeps going but they tell us to round to the nearest Isles it says zero point nine seven seven zero point nine seven seven 0.977 rounded to the nearest thousandth and then they finally say how many becquerels of caesium-137 remain in our sample 150 days 150 days after its release in the soil use the rounded value of our and round this number to the nearest hundredth so just to be clear we now have our we already know C and R we know that the amount of caesium-137 in becquerels as a function of time and days is going to be equal to eight times zero point nine seven seven to the T power where T is the number of days that have passed and they're essentially saying well how much do we have left after 150 days so they want us to calculate this what is a of 150 well that's going to be eight times zero point nine seven seven to the 150th how to the 150th power and so clearly we need a calculator for this so let's calculate that so it's going to be eight times and they told us to use our rounded value of r-not the exact value of our so it's going to be eight times zero point nine seven seven to the 150th power and that gets us point and they want us to round to the nearest hundredth zero point two four zero point zero point two four is in zero point two four becquerels is kind of the radioactivity level the caesium-137 that we have left over now one interesting thing is they asked us to use the rounded value of R so we use the rounded value of R but you could because this right over here is a multiple of 30 you could actually not too difficult away find out the exact value that's left over any actually you don't even need a calculator for it I encourage you to pause your video and try to think about that find the exact value well instead of writing 0.977 let's write a of T as being equal to 8 times R R this is an approximate value for R if we wanted to be a little more exact we can say that our R is 1/2 to the one thirtieth power and we're going to raise that we're going to raise that to the T power or we could say a of T is equal to 8 times 1/2 to the T over 30 power if we raise something to an exponent then raise that to an exponent we can take the product of those exponents so that's 1/2 to the T over 30 power let me actually do that in another color I'm doing yellow so that's 8 times 1/2 to the one-half to the T over 30 power 1/2 to the T over 30 power and I don't need this I don't need this parenthesis right over here this is another way to describe a of T so what is a of 150 so a of 150 is going to be equal to 8 times 1/2 1/2 to the 1 50 over 30 well that's just 5 1/2 to the fifth power well what's 1/2 to the fifth power that is 1 to the fifth over 2 to the fifth or 1 over 32 so this right over here is 1 over 32 which is equal to 8 over 32 8 over 32 which is equal to 1 over 4 which is equal to 1 over 4 or 0.25 or 0.25 so are using our approximation for our we got zero point two four when we rounded to the 150th power so that's using our approximation a lot where buying we're taking 150 of these and multiplying by multiplying them together but it's not too far off of what the real value is and they asked us to use the rounded value but if we'd use the precise value the actual value we would have gotten Oh point 0.25 becquerels would be left over