Exponential models: half-life and doubling time

About this lesson

Precalculus › Exponential and logarithmic functions

To save a PDF, choose “Save as PDF” in the print dialog.

Exponential models: half-life and doubling time

Warm-up

Three quick questions from earlier lessons.

Discover

A doctor gives a patient 80 mg of a medicine. Every 6 hours, half of what is left leaves the body.

How much is left after 6 hours? After 12? After 24? Does the amount ever reach 0?

Key idea

The half-life hh is the time it takes for an amount to halve; the doubling time dd is the time it takes to double.

Half-life hh: A(t)=A0(12)thA(t) = A_0 \left(\tfrac{1}{2}\right)^{\frac{t}{h}}. Doubling time dd: P(t)=P0⋅2tdP(t) = P_0 \cdot 2^{\frac{t}{d}}

The exponent th\frac{t}{h} counts how many half-lives have passed.

Worked example

A sample of 80 mg has a half-life of 5 years. How much is left after 15 years?

155=3\dfrac{15}{5} = 3 half-lives
Count the half-lives.
A=80(12)3A = 80 \left(\tfrac{1}{2}\right)^3
Halve three times.
A=10A = 10 mg
80 ÷ 8 = 10.

Your turn, guided

Your turn: 96 mg, half-life 2 hours. How much is left after 8 hours? Fill in the missing line.

A=96(12)4A = 96 \left(\tfrac{1}{2}\right)^4
8 ÷ 2 = 4 half-lives.
A=6A = 6 mg
96 ÷ 16 = 6.

Practice ★

  1. A sample of 64 mg of a radioactive substance has a half-life of 2 years. How many mg are left after 6 years?
  2. A sample of 64 mg of a radioactive substance has a half-life of 3 years. How many mg are left after 9 years?
  3. A sample of 96 mg of a radioactive substance has a half-life of 5 years. How many mg are left after 20 years?
  4. A sample of 320 mg of a radioactive substance has a half-life of 10 years. How many mg are left after 40 years?
  5. A sample of 320 mg of a radioactive substance has a half-life of 2 years. How many mg are left after 10 years?

Watch out!

Practice ★★

  1. A sample of 200 mg has a half-life of 2 years. How much is left after 8 years? Round to 2 decimal places.
  2. A sample of 100 mg has a half-life of 3 years. How much is left after 11 years? Round to 2 decimal places.
  3. A sample of 400 mg has a half-life of 3 years. How much is left after 4 years? Round to 2 decimal places.
  4. A sample of 200 mg has a half-life of 3 years. How much is left after 10 years? Round to 2 decimal places.
  5. A sample of 100 mg has a half-life of 4 years. How much is left after 11 years? Round to 2 decimal places.

Exit ticket

Three questions to check you have got it.

  1. A dose of 200 mg of a medicine has a half-life of 2 hours. Write the amount A(t)A(t), in mg, left after tt hours.
  2. A dose of 50 mg of a medicine has a half-life of 3 hours. Write the amount A(t)A(t), in mg, left after tt hours.
  3. A dose of 500 mg of a medicine has a half-life of 4 hours. Write the amount A(t)A(t), in mg, left after tt hours.

Summary card

Half-life and doubling time

  • Half-life hh: A0(12)thA_0 \left(\tfrac{1}{2}\right)^{\frac{t}{h}}. Doubling time dd: P0⋅2tdP_0 \cdot 2^{\frac{t}{d}}.
  • th\frac{t}{h} = number of half-lives.

Watch out: nn half-lives divide by 2n2^n, not by 2n2n.

End of the lesson

Next lesson: Properties of logarithms: product, quotient and power rules Back to the chapter

Answers: Exponential models: half-life and doubling time

A

  1. 88
  2. 88
  3. 66
  4. 2020
  5. 1010

B

  1. 252\frac{25}{2} mg
  2. 25234\frac{25 \sqrt[3]{2}}{4} mg
  3. 100⋅223100 \cdot 2^{\frac{2}{3}} mg
  4. 25⋅2232\frac{25 \cdot 2^{\frac{2}{3}}}{2} mg
  5. 25242\frac{25 \sqrt[4]{2}}{2} mg

C

  1. 200⋅2−t2200 \cdot 2^{- \frac{t}{2}}
  2. 50⋅2−t350 \cdot 2^{- \frac{t}{3}}
  3. 500⋅2−t4500 \cdot 2^{- \frac{t}{4}}

CC BY-SA 4.0