Half-Life Problems

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Show your work for each problem.

1.
A substance has a half-life of $10$ years. What fraction of it is left after $2$ half-lives?
1. a) $0$
2. b) $\frac{1}{4}$
3. c) $\frac{1}{2}$
4. d) $\frac{1}{8}$
2.
A sample of $60$ g has a half-life of $5$ days. How many grams are left after $5$ days?
3.
A sample starts with $50$ mg and has a half-life of $3$ hours. Which function gives the amount $N(t)$ after $t$ hours?
1. a) $N(t) = 50 \left(\frac{1}{2}\right)^{3t}$
2. b) $N(t) = 3 \left(\frac{1}{2}\right)^{t/50}$
3. c) $N(t) = 50 \left(\frac{1}{2}\right)^{t/3}$
4. d) $N(t) = 50 - \frac{1}{2} t$
4.
A medicine sample has $320$ mg and a half-life of $2$ hours. How many mg are left after $6$ hours?
5.
Iodine-131 has a half-life of $8$ days. How many half-lives pass in $24$ days?
6.
What percent of a radioactive sample is left after $3$ half-lives? (Give the percent as a number, for example 50 for $50\%$ .)
7.
A sample has $\frac{1}{16}$ of its original amount left. How many half-lives have passed?
1. a)8
2. b)16
3. c)4
4. d)2
8.
A wooden tool contains $\frac{1}{4}$ of its original carbon-14. The half-life of carbon-14 is $5730$ years. How many years old is the tool?
9.
A sample of $240$ g decays to $30$ g in $18$ hours. What is its half-life in hours?
10.
A hospital has $500$ mg of iodine-131 (half-life $8$ days). How many mg are left after $40$ days?
11.
A student says: a $100$ g sample loses $50$ g every half-life, so after $2$ half-lives nothing is left. How much is really left after $2$ half-lives?
1. a)0 g
2. b)50 g
3. c)75 g
4. d)25 g
12.
A patient takes $800$ mg of a medicine with a half-life of $5$ hours. How many mg are left after $20$ hours?
13.
A substance has a half-life of $3$ days. After how many days is only $\frac{1}{32}$ of it left?
14.
What percent of a radioactive sample has decayed (is gone) after $3$ half-lives?
1. a)37.5%
2. b)75%
3. c)87.5%
4. d)12.5%
15.
A patient takes $800$ mg of a medicine with a half-life of $4$ hours. How many mg are left after $6$ hours? Use $\left(\frac{1}{2}\right)^{1.5} \approx 0.3536$ and round to the nearest mg.
16.
A fossil contains $\frac{1}{64}$ of its original carbon-14. The half-life of carbon-14 is $5730$ years. How many years old is the fossil?
17.
A $2000$ g sample decays to $250$ g in $21$ days. Find the half-life, then find how many grams are left after $35$ days (from the start).
18.
A patient takes $300$ mg of a medicine with a half-life of $4$ hours. The medicine counts as cleared when less than $10$ mg is left. After how many hours (a whole number of half-lives) is that first true?