Q1. The half-life of an isotope is 24 hours. If 3 milligrams are present now, how much will be present in 72 hours?

Background

Topic: Exponential Decay and Half-Life

This question tests your understanding of exponential decay, specifically using the half-life formula to determine the remaining quantity of a substance after a given time.

Key Terms and Formulas

• Half-life (): The time required for a quantity to reduce to half its initial value.

• Exponential Decay Formula:

• Where:

  • = amount remaining after time

  • = initial amount

  • = elapsed time

  • = half-life

Step-by-Step Guidance

1. Identify the known values: mg, hours, hours.

2. Plug these values into the exponential decay formula:

3. Simplify the exponent:

4. Rewrite the expression:

Try solving on your own before revealing the answer!

Q2. A scientist has a 4-g sample of a radioactive material with a half-life of 10 hours. How old is the second (in hours) if 0.5 g remains?

Background

Topic: Exponential Decay and Solving for Time

This question tests your ability to use the half-life formula to solve for the elapsed time when given the initial and remaining amounts.

Key Terms and Formulas

• Exponential Decay Formula:

• To solve for , rearrange:

• Take the logarithm of both sides to solve for .

Step-by-Step Guidance

1. Identify the known values: g, g, hours.

2. Set up the equation:

3. Divide both sides by 4:

4. Simplify:

5. Take the logarithm of both sides to solve for .

Try solving on your own before revealing the answer!

Q3. The logistic growth model represents the population of a bacteria in a culture tube after hours. What is the number of bacteria in 4 hours?

Background

Topic: Logistic Growth Model

This question tests your ability to evaluate a logistic growth function at a specific time to find the population.

Key Terms and Formulas

• Logistic Growth Model:

• Where:

  • = carrying capacity

  • = constant determined by initial conditions

  • = growth rate

  • = time

Step-by-Step Guidance

1. Identify the values: , , , .

2. Plug into the formula:

3. Calculate the exponent:

4. Rewrite the denominator:

5. Set up the expression for to evaluate.

Try solving on your own before revealing the answer!

Q4. A thermometer reading 70°F is placed in a cold storage room with a constant temperature of 40°F. If the thermometer reads 58°F after 2 minutes, how long before it will read 51°F? Assume the cooling follows Newton’s Law of Cooling:

Background

Topic: Newton’s Law of Cooling

This question tests your ability to use Newton’s Law of Cooling to solve for the time required for an object to reach a certain temperature.

Key Terms and Formulas

• Newton’s Law of Cooling:

• Where:

  • = temperature at time

  • = surrounding temperature

  • = initial temperature

  • = cooling constant (negative)

  • = time

Step-by-Step Guidance

1. Identify the known values: , , .

2. First, use the information at to solve for :

3. Set up the equation:

4. Solve for (do not compute the final value yet).

5. Next, use to set up the equation for when and solve for .

Try solving on your own before revealing the answer!