smartcookie1/2thetime
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- Jan 28, 2021
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An isotope of Caesium has a half-life of 30 years, which means it loses 50% of its atoms every 30 years. Suppose we start with 10 kg sample of Caesium.
Q1. Write a rule that will calculate the amount of Caesium after ANY number of years.
So first I figured that if you begin with 10 kg, that would be the beginning part of the equation. Then after that I figured, if it loses 50% then you would do 100-50 which is just 50, or 0.5. And then you'd square that to get the half-life. This is what I have:
f(x)= 10*0.5^x
But I know that this isn't correct because when you plug in a given number, say 30 you'd expect the answer to be 5, because half of 10 is 5. Instead though, I got this 9.313 number, which is nowhere near close. While I'm doing this, I'm talking to a friend, and apparently my x is in half lives instead of years, which is a problem, because I am supposed to be figuring out the amount Caesium. And then he says that I instead I should be looking for the number that multiplies by 30 times to get to 0.5. Which basically means you have to find the root number, I think. Like how the 2 root of 16 is 4^2 or 4*4. At first I was like, okay, divided 0.5 by 30, but I realize it wouldn't work like that because that would be the opposite of multiplying. After that I was thinking maybe square rooting, but that doesn't work either.
And this is where I am stuck. Hopefully, this makes as much sense in my head as it does on paper. So I guess what my question is, how do I find the root number without having to spend and obnoxious amount of time plugging in a bunch of random numbers.
Q1. Write a rule that will calculate the amount of Caesium after ANY number of years.
So first I figured that if you begin with 10 kg, that would be the beginning part of the equation. Then after that I figured, if it loses 50% then you would do 100-50 which is just 50, or 0.5. And then you'd square that to get the half-life. This is what I have:
f(x)= 10*0.5^x
But I know that this isn't correct because when you plug in a given number, say 30 you'd expect the answer to be 5, because half of 10 is 5. Instead though, I got this 9.313 number, which is nowhere near close. While I'm doing this, I'm talking to a friend, and apparently my x is in half lives instead of years, which is a problem, because I am supposed to be figuring out the amount Caesium. And then he says that I instead I should be looking for the number that multiplies by 30 times to get to 0.5. Which basically means you have to find the root number, I think. Like how the 2 root of 16 is 4^2 or 4*4. At first I was like, okay, divided 0.5 by 30, but I realize it wouldn't work like that because that would be the opposite of multiplying. After that I was thinking maybe square rooting, but that doesn't work either.
And this is where I am stuck. Hopefully, this makes as much sense in my head as it does on paper. So I guess what my question is, how do I find the root number without having to spend and obnoxious amount of time plugging in a bunch of random numbers.