Problem 42 - Intermediate Algebra

Question

The number of grams \(Q\) of a certain radioactive substance present after \(t\) seconds is given by the equation \(Q=\) \(1500 e^{-0.4 t}\). How many grams remain after 5 seconds? 10 seconds? 20 seconds?

Step-by-Step Solution

Answer

After 5 seconds: ~203 grams; after 10 seconds: ~27 grams; after 20 seconds: ~0.5 grams.

Step 1: Understand the Problem

The equation given is \( Q = 1500 e^{-0.4t} \), which calculates the amount \( Q \) of a radioactive substance in grams after \( t \) seconds. We need to find \( Q \) for \( t = 5, 10, \) and \( 20 \) seconds.

Step 2: Substitute and Calculate for 5 Seconds

Substitute \( t = 5 \) into the equation \( Q = 1500 e^{-0.4t} \). This gives \( Q = 1500 e^{-0.4 imes 5} = 1500 e^{-2} \). Calculate \( Q \) using the value of \( e^{-2} \).

Step 3: Calculate Exponential and Result for 5 Seconds

The value of \( e^{-2} \) is approximately 0.1353. Multiply this by 1500: \( Q = 1500 imes 0.1353 = 202.95 \). So, after 5 seconds, about 203 grams of the substance remain.

Step 4: Substitute and Calculate for 10 Seconds

Substitute \( t = 10 \) into the equation. Thus, \( Q = 1500 e^{-0.4 imes 10} = 1500 e^{-4} \). Calculate \( Q \) using the value of \( e^{-4} \).

Step 5: Calculate Exponential and Result for 10 Seconds

The value of \( e^{-4} \) is approximately 0.0183. Multiply this by 1500: \( Q = 1500 imes 0.0183 = 27.45 \). So, after 10 seconds, about 27 grams of the substance remain.

Step 6: Substitute and Calculate for 20 Seconds

Substitute \( t = 20 \) into the equation. Thus, \( Q = 1500 e^{-0.4 imes 20} = 1500 e^{-8} \). Calculate \( Q \) using the value of \( e^{-8} \).

Step 7: Calculate Exponential and Result for 20 Seconds

The value of \( e^{-8} \) is approximately 0.000335. Multiply this by 1500: \( Q = 1500 imes 0.000335 = 0.5025 \). So, after 20 seconds, about 0.5 grams of the substance remain.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process in which unstable atomic nuclei lose energy by emitting radiation. Over time, this causes the substance to change into a different element or isotope.
It is measured using a specific rate known as the decay constant.

• Radioactive substances naturally decay at an exponential rate.
• The decay of a radioactive element can be modeled by an exponential decay equation, like the one shown in our exercise: \( Q = 1500 e^{-0.4t} \).
• This equation helps predict how much of a radioactive substance will remain after a certain amount of time.

Understanding the decay equation allows us to calculate the remaining mass of a substance at different time intervals.
This is crucial for fields such as archaeology, medicine, and nuclear energy, where gauging time or dosage is essential.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They often model real-world situations where things either grow or shrink rapidly.

• In the formula \( Q = 1500 e^{-0.4t} \), \( e \) is the base of the natural logarithm, and the exponent is \(-0.4t\).
• An exponential decay function decreases over time, as seen by the negative exponent in our equation.
• The base \( e \) (approximately 2.718) is a crucial component of continuous growth and decay equations.

The power of exponential functions lies in their ability to describe rapid changes.
In real-life scenarios, they predict population growth, the spread of diseases, and the depreciation of assets.

Mathematical Modeling

Mathematical modeling involves creating abstract representations of a real-world scenario using mathematical language and symbols. It helps predict and analyze the behavior of systems.
In our exercise, the equation \( Q = 1500 e^{-0.4t} \) models the decay of a radioactive substance.
A few key points about mathematical modeling include:

• Models can simplify complex realities and aid in understanding potential outcomes.
• They rely on assumptions and variables that must be carefully chosen and validated.
• Mathematical models are essential in scientific research and technology development.

By applying this decay model, scientists can estimate how the radioactive material breaks down over time.
It becomes an invaluable tool in planning and decision-making in various industries.