Half-Life Problems

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) $\frac{1}{8}$
Answer: $\frac{1}{4}$
After $2$ half-lives the fraction left is $\left(\frac{1}{2}\right)^{2} = \frac{1}{4}$ . It is not $0$ : each half-life removes only half of what is left.
2.
A sample of $60$ g has a half-life of $5$ days. How many grams are left after $5$ days?
Answer: 30
$5$ days is $1$ half-life, so the sample is halved once: $60 \times \frac{1}{2} = 30$ g.
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) $N(t) = 50 - \frac{1}{2} t$
Answer: $N(t) = 50 \left(\frac{1}{2}\right)^{t/3}$
$N_0 = 50$ and $T = 3$ , so $N(t) = 50 \left(\frac{1}{2}\right)^{t/3}$ . The exponent $\frac{t}{3}$ counts the half-lives. The last option is linear, not exponential.
4.
A medicine sample has $320$ mg and a half-life of $2$ hours. How many mg are left after $6$ hours?
Answer: 40
1. How many half-lives are in 6 hours? 3
2. What is $2^3$ ? 8
3. What is $320 \div 8$ ? 40
5.
Iodine-131 has a half-life of $8$ days. How many half-lives pass in $24$ days?
Answer: 3
$\frac{t}{T} = \frac{24}{8} = 3$ half-lives.
6.
What percent of a radioactive sample is left after $3$ half-lives? (Give the percent as a number, for example 50 for $50\%$ .)
Answer: 12.5
1. What fraction is left? (Write it like 1/2.) 1/8
2. Write 1/8 as a decimal. 0.125
3. Write 0.125 as a percent (number only). 12.5
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
Answer: 4
$\frac{1}{16} = \frac{1}{2^4} = \left(\frac{1}{2}\right)^{4}$ , so $4$ half-lives have passed ( $1 \to \frac{1}{2} \to \frac{1}{4} \to \frac{1}{8} \to \frac{1}{16}$ ). The denominator 16 is not the number of half-lives.
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?
Answer: 11460
$\frac{1}{4}$ left means $2$ half-lives, so the age is $2 \times 5730 = 11460$ years.
9.
A sample of $240$ g decays to $30$ g in $18$ hours. What is its half-life in hours?
Answer: 6
1. How many times smaller is the sample? ( $240 \div 30$ ) 8
2. $8 = 2^n$ . What is $n$ (the number of half-lives)? 3
3. What is the half-life in hours? 6
10.
A hospital has $500$ mg of iodine-131 (half-life $8$ days). How many mg are left after $40$ days?
Answer: 15.625
1. How many half-lives are 40 days? 5
2. What is $2^5$ ? 32
3. What is $500 \div 32$ ? (exact decimal) 15.625
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
Answer: 25 g
$100 \to 50 \to 25$ : the second half-life removes half of $50$ , not $50$ g. So $25$ g is left.
12.
A patient takes $800$ mg of a medicine with a half-life of $5$ hours. How many mg are left after $20$ hours?
Answer: 50
$\frac{20}{5} = 4$ half-lives: $800 \left(\frac{1}{2}\right)^{4} = \frac{800}{16} = 50$ mg ( $800 \to 400 \to 200 \to 100 \to 50$ ).
13.
A substance has a half-life of $3$ days. After how many days is only $\frac{1}{32}$ of it left?
Answer: 15
1. $32 = 2^n$ . What is $n$ (the number of half-lives)? 5
2. How many days are 5 half-lives? 15
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%
Answer: 87.5%
Left: $\frac{1}{8} = 12.5\%$ . Decayed: $1 - \frac{1}{8} = \frac{7}{8} = 87.5\%$ . The answer $12.5\%$ is the part that is still there.
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.
Answer: 283
1. How many half-lives are 6 hours? ( $\frac{t}{T}$ ) 1.5
2. Compute $800 \times 0.3536$ . 282.88
3. Round to the nearest mg. 283
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?
Answer: 34380
$64 = 2^6$ , so $6$ half-lives have passed: $6 \times 5730 = 34380$ years.
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).
Answer: 62.5
1. $2000 \div 250 = 8 = 2^n$ . What is $n$ ? 3
2. What is the half-life in days? 7
3. How many half-lives are 35 days? 5
4. What is $2000 \div 2^5$ ? 62.5
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?
Answer: 20
1. How many mg are left after 4 half-lives? ( $300 \div 16$ ) 18.75
2. How many mg are left after 5 half-lives? 9.375
3. How many half-lives until less than 10 mg is left? 5
4. How many hours is that? 20