The half-life of cesium-137 is 30 years. Suppose we have a
100-mg sample.
(a) Find the mass that remains after t years.
(b) How much of the sample remains after 100 years?
(c) What is the rate of decay after 100 years?
(d) After how long will only 1 mg remain?

Answer

a) $$
m(t)=100 \times(2)^{-\frac{t}{30}}
$$
b) $$
9.92 \mathrm{mg}
$$
c) $$
199.3
$$
d) approximately after $[199.3]$ years, the sample will remain only 1 $\mathrm{mg}$

Related Courses

Calculus 1 / AB

Biocalculus Calculus for the Life Sciences

Chapter 3

Derivatives

Section 6

Exponential Growth and Decay

Discussion

You must be signed in to discuss.

Top Calculus 1 / AB Educators

Watch More Solved Questions in Chapter 3

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Video Transcript

now for problem on Percival. It's they I m f t If the mess off season in Medic Rams remaining after time t in years, so am with T is equal to a most zero. The phone message season at years here, multiplied by exponential Okay, key. So Mosul is giving to be 100 milligrams exponential k g. Now you need to get the value of K So it's that kink it the message season at any year t so first, to get the value off K, we have another giving that the half life of caesium is 30 years. So a month 30 is actually equal toe have off him of deal. So I am a 30 is equal to 100 exponential Okay, marked by by search losing this two room and the half life cesium is have multiplied by 100. Oh, we can terminate this with 100 with this 100. So we have hey 13 d equal toe 1/2 and we can't take then both sides. So here we will have 30 k equals to lend Hey and let half is also equal toe negative Len tool for simplicity. So the K value would be negative men to over 30. And now we can say that I am with T is the mess it in here is equal to 100 exponential. Negative. Then toe over 30 Gay T shirt over certainty. Okay. And this can be simplified that a mufti is equal toe 2200. Sorry, 200 with the power off Negative t over third cause exponential lento is equal to to So here we have 200. So now we have got that The main function so was part B. We want I don't know after 20 is equal to 100 How much mace remains So we basically saying am off 100 tired Substitution is equal toe 200 with the dollar off negative 100 over search which is equal pool about 9.92 Many grapes now was part See you will try to get the rate of decay So we will start by using the for the functions that we have got Wife he's equal. Do we will have 20 200 Sorry. Five t equals 200 through the park. Negative t over 30. So why dish 50 which is a decay over time. At any year, T is equal to 200 with the power of negative T over 30 multiplied by negative one over 13 Let me to. And now at year 100 Why dash 100? The rate of decay exactly as a year. 100 is equal toe 100 Oh, I'm sorry. Is equal to 200 with the power of negative 100 over 30 multiplied by negative one over 30 thing to which is equal to negative Whole point to to nine milligram were you that is only logical is the negative value your prisons that it is a decayed It is a losing mess And this value is how much mess will be lost at year 100. So this is great. How much milligrams when the cesium lose at year Betty and this And this street is at your 100 now for part d you want to get when with why 50? When will it be equal? Only one milligram. So we will move like backward. Get the value or five t that is equal toe one milligram. So we will say that 100 But we will take one step back will be much easier. 100 e with the power of negative lento over 30 g. So instead of using the value of 100 we took a step back using this exponential off negative then and that is equal No. One. It's divide both sides by 100 giving us e to the port of negative lento over 30 de equal cool, one over 100 and we will take win for both sides. So here we have negative Lynn over 30 de equal tool Lynn one over 100. And now we can get value can get the value of T the year at which decision will be one milligram and that is equal to year 100 99 went three. So it will take decision around two centuries to decay from 100 milligram toe, only one milligram. Thank you

Get More Help with this Textbook