Lecture 13 - Calculus Study Guide: Doubling Time and Half-Life Problems

Q1. Disease Doubling: How many infected people are there t days after March 1, if the number doubles every 3 days and there were 100 infected people on March 1?

Background

Topic: Exponential Growth and Doubling Time

This question tests your understanding of exponential growth, specifically using the doubling time formula to model population growth over time.

Key Terms and Formulas

• Doubling Time (K): The period it takes for a quantity to double in size.

• Exponential Growth Formula (Doubling):

$\text{Amount after } t \text{ minutes} = A \times 2^{t/K}$

• $A$ = initial amount

• $K$ = doubling time

• $t$ = time elapsed

Step-by-Step Guidance

1. Identify the initial amount $A = 100$ (infected people on March 1).

2. Recognize the doubling time $K = 3$ days.

3. Write the exponential growth formula for this scenario: $\text{Number of infected after } t \text{ days} = 100 \times 2^{t/3}$

4. To find the number of infected people after $t$ days, substitute the value of $t$ into the formula above.

Try solving on your own before revealing the answer!

Q2. Disease Doubling: How many days until there are 1000 infected people?

Background

Topic: Solving Exponential Equations for Time

This question asks you to solve for the time required for an exponentially growing population to reach a certain size, given the doubling time.

Key Terms and Formulas

• Use the same exponential growth formula: $\text{Amount} = A \times 2^{t/K}$

• To solve for $t$, you'll need to use logarithms.

Step-by-Step Guidance

1. Set up the equation: $1000 = 100 \times 2^{t/3}$

2. Divide both sides by 100 to isolate the exponential term: $10 = 2^{t/3}$

3. Take the logarithm of both sides (base 2 is most direct): $\log_2(10) = t/3$

4. Solve for $t$ by multiplying both sides by 3: $t = 3 \times \log_2(10)$

Try solving on your own before revealing the answer!

Q3. Mold Growth: What is the doubling time (in days) if a mold colony grows from 10 mg to 30 mg in one week?

Background

Topic: Finding Doubling Time from Exponential Growth Data

This question tests your ability to determine the doubling time when given two measurements at different times.

Key Terms and Formulas

• Exponential growth formula: $\text{Final amount} = \text{Initial amount} \times 2^{t/K}$

• Here, $t$ is the elapsed time (in days), $K$ is the doubling time (in days).

Step-by-Step Guidance

1. Let $A = 10$ mg (initial), final amount $= 30$ mg, and $t = 7$ days (one week).

2. Set up the equation: $30 = 10 \times 2^{7/K}$

3. Divide both sides by 10: $3 = 2^{7/K}$

4. Take the logarithm of both sides (base 2): $\log_2(3) = 7/K$

5. Rearrange to solve for $K$: $K = 7/\log_2(3)$

Try solving on your own before revealing the answer!

Q4. Radioactive Decay: Isotope W has a half-life of 10 years. How much remains after 20 years?

Background

Topic: Exponential Decay and Half-Life

This question tests your understanding of exponential decay and how to use the half-life formula to determine the remaining amount of a substance after a given time.

Key Terms and Formulas

• Half-life (H): Time required for half of a substance to decay.

• Exponential decay formula:

$\text{Amount remaining} = A \times (1/2)^{t/H}$

• $A$ = initial amount

• $H$ = half-life

• $t$ = time elapsed

Step-by-Step Guidance

1. Let $A$ be the initial amount (not specified, so keep as $A$).

2. Half-life $H = 10$ years, $t = 20$ years.

3. Plug into the formula: $\text{Amount remaining} = A \times (1/2)^{20/10}$

4. Simplify the exponent: $20/10 = 2$, so $\text{Amount remaining} = A \times (1/2)^2$

Try solving on your own before revealing the answer!

Q5. Radioactive Decay: If there are initially 80 grams of a substance with a half-life of 7 years, how much remains after 14 years?

Background

Topic: Exponential Decay and Half-Life

This question is similar to the previous one, but with specific values for the initial amount and half-life.

Key Terms and Formulas

• Exponential decay formula: $\text{Amount remaining} = A \times (1/2)^{t/H}$

Step-by-Step Guidance

1. Initial amount $A = 80$ grams, half-life $H = 7$ years, $t = 14$ years.

2. Plug into the formula: $\text{Amount remaining} = 80 \times (1/2)^{14/7}$

3. Simplify the exponent: $14/7 = 2$, so $\text{Amount remaining} = 80 \times (1/2)^2$

Try solving on your own before revealing the answer!

Q6. Radioactive Decay: If there are initially 70 grams of a substance with a half-life of 5 years, how much remains after t years?

Background

Topic: Exponential Decay and Half-Life (General Formula)

This question asks you to write a general formula for the remaining amount after $t$ years, given the initial amount and half-life.

Key Terms and Formulas

• Exponential decay formula: $\text{Amount remaining} = A \times (1/2)^{t/H}$

Step-by-Step Guidance

1. Initial amount $A = 70$ grams, half-life $H = 5$ years.

2. Plug into the formula: $\text{Amount remaining} = 70 \times (1/2)^{t/5}$

3. This formula gives the remaining amount for any value of $t$.

Try solving on your own before revealing the answer!

Q7. Radioactive Decay: How many years until 10 grams remain (starting from 70 grams, half-life 5 years)?

Background

Topic: Solving for Time in Exponential Decay

This question asks you to solve for the time required for a substance to decay to a certain amount, given the initial amount and half-life.

Key Terms and Formulas

• Exponential decay formula: $\text{Amount remaining} = A \times (1/2)^{t/H}$

• To solve for $t$, use logarithms.

Step-by-Step Guidance

1. Set up the equation: $10 = 70 \times (1/2)^{t/5}$

2. Divide both sides by 70: $10/70 = (1/2)^{t/5}$

3. Take the logarithm of both sides (base 1/2): $\log_{1/2}(10/70) = t/5$

4. Solve for $t$ by multiplying both sides by 5: $t = 5 \times \log_{1/2}(10/70)$

Try solving on your own before revealing the answer!

Q8. Carbon-14 Dating: A bone is found with 2% of the usual amount of carbon-14. The half-life of carbon-14 is 5730 years. How old is the bone?

Background

Topic: Radioactive Decay and Carbon Dating

This question tests your ability to use the half-life formula to determine the age of an object based on the remaining percentage of a radioactive isotope.

Key Terms and Formulas

• Exponential decay formula: $\text{Amount remaining} = A \times (1/2)^{t/H}$

• Here, $A$ is the original amount, $t$ is the time elapsed, $H$ is the half-life.

Step-by-Step Guidance

1. Let $A$ be the original amount, and the remaining amount is $0.02A$ (2%).

2. Set up the equation: $0.02A = A \times (1/2)^{t/5730}$

3. Divide both sides by $A$: $0.02 = (1/2)^{t/5730}$

4. Take the logarithm of both sides (base 1/2): $\log_{1/2}(0.02) = t/5730$

5. Solve for $t$ by multiplying both sides by 5730: $t = 5730 \times \log_{1/2}(0.02)$

Try solving on your own before revealing the answer!