Problem 67 - College Algebra and Calculus: An Applied Approach

Question

Radioactive Decay Strontium-90 has a half-life of \(29.1\) years. The amount \(S\) of 100 kilograms of strontium-90 present after \(t\) years is given by \(S=100 e^{-0.0238 t}\) How much of the 100 kilograms will remain after 50 years?

Step-by-Step Solution

Answer

Based on the calculation, approximately \(29.5\) kilograms of Strontium-90 will remain after 50 years.

Step 1: Understand the Decay Formula

In the formula \(S=100e^{-0.0238t}\), \(S\) represents the remaining mass of strontium-90, while \(t\) represents the time passed in years. The number \(-0.0238\) is the decay constant, which is calculated using the half-life of the radioactive substance.

Step 2: Substituting the values

We need to find the amount of Strontium-90 remaining after 50 years. Thus, we will substitute \(t = 50\) into the formula: \(S=100e^{-0.0238*50}\).

Step 3: Calculate the Remaining Amount

To solve for \(S\), we calculate the exponentiation of \(e\) and the multiplication first, and then multiply the result by 100.

Key Concepts

Exponential Functions

Exponential functions often appear in the context of growth and decay processes. They can describe how quantities grow or shrink over time. This is especially relevant in radioactive decay, where substances decrease in quantity at a rate proportional to their current value.
The general form of an exponential function is given by\[ y = ab^{x} \] where:

• \( a \) is the initial amount,
• \( b \) is the base of the exponential, which determines the growth (if \( b > 1 \)) or decay (if \( 0 < b < 1 \)) rate,
• \( x \) is the exponent, representing time or another independent variable.

For continuous decay, such as radioactive decay, the function adopts the form \[ y = ae^{-kx} \]where \( e^{ } \) is the base of the natural logarithm, and \( k \) is the decay constant. This results in a smooth gradual decrease of the substance, in contrast to linear or abrupt changes.

Half-Life Calculations

Half-life is a crucial concept in understanding radioactive decay. It refers to the amount of time it takes for half of the radioactive material to decay. This period remains constant and is specific to each radioactive substance.
To compute the half-life, the following formula is used:\[ t_{1/2} = \frac{0.693}{k} \]where:

• \( t_{1/2} \) is the half-life,
• \( k \) is the decay constant.

For Strontium-90, with a half-life of 29.1 years, we can understand how this affects the quantity of the substance over time. After each half-life period, only half of the original amount remains.
This concept helps estimate how long a radioactive substance will persist and aids in planning and safety considerations in fields like nuclear medicine and waste management.

Decay Constant

The decay constant is a pivotal factor in the mathematics of radioactive decay. It directly affects how quickly a substance loses its radioactive properties.
In the decay formula, such as \[ S = ae^{-kt} \] the decay constant \( k \) determines the rate at which the exponential function decreases over time. A higher decay constant indicates a faster decay, meaning the substance will reach half of its original amount more quickly.
The decay constant is related to the half-life by the formula \[ k = \frac{0.693}{t_{1/2}} \] When solving problems, this relation is often used to calculate one value when the other and the half-life are known. It's an essential parameter in the safe handling and utilization of radioactive materials.