Problem 81 - Algebra and Trigonometry

Question

A 15 -g sample of radioactive iodine decays in such a way that the mass remaining after \(t\) days is given by \(m(t)=15 e^{-0.087 t},\) where \(m(t)\) is measured in grams. After how many days is there only 5 g remaining?

Step-by-Step Solution

Answer

It takes about 12.61 days for the mass to reduce to 5 grams.

Step 1: Understanding the Problem

We need to find after how many days the mass of radioactive iodine will decrease to 5 grams. We are given the function for remaining mass: \( m(t) = 15 e^{-0.087t} \).

Step 2: Setting Up the Equation

To find the time, \( t \), when the mass is 5 grams, we set \( m(t) = 5 \). This gives us the equation: \( 5 = 15 e^{-0.087t} \).

Step 3: Isolating the Exponential

Divide both sides of the equation by 15 to isolate the exponential part: \( \frac{5}{15} = e^{-0.087t} \), which simplifies to \( \frac{1}{3} = e^{-0.087t} \).

Step 4: Applying the Natural Logarithm

Apply the natural logarithm to both sides to solve for \( t \): \( \ln\left(\frac{1}{3}\right) = \ln\left(e^{-0.087t}\right) \).

Step 5: Simplifying the Logarithm Equation

Use the property \( \ln(e^x) = x \) to simplify: \( \ln\left(\frac{1}{3}\right) = -0.087t \).

Step 6: Solving for Time \( t \)

Solve for \( t \) by dividing both sides by \(-0.087\): \( t = \frac{\ln\left(\frac{1}{3}\right)}{-0.087} \).

Step 7: Calculating the Result

The value of \( t \) is approximately \( t \approx \frac{\ln(0.3333)}{-0.087} \). Using a calculator gives \( t \approx 12.61 \). Therefore, it takes approximately 12.61 days for the mass to be 5 grams.

Key Concepts

Exponential Functions

Exponential functions are a fascinating concept in mathematics, and they often appear in real-world applications like radioactive decay. In the given problem of radioactive iodine, the function that represents the changing mass over time is exponential: \[ m(t) = 15 e^{-0.087t} \] The letter \(e\) here stands for Euler's number, which is approximately 2.718 and is a constant that appears frequently in mathematics. Exponential functions have the general form of \(f(t) = ab^{t}\), where \(a\) is the initial value, \(b\) is the base of the exponential, and \(t\) usually represents time. For radioactive decay, we adjust the formula using Euler's number to better model continuous decay. In this function:

• 15 is the initial mass of the iodine sample in grams,
• \(e^{-0.087t}\) represents how the mass decreases over time,
• and the exponent \(-0.087t\) indicates the decay’s speed.

The negative sign in the exponent \(-0.087t\) confirms that this is a decay function, because the mass is continuously decreasing.

Natural Logarithm

A natural logarithm is an essential tool for solving exponential equations like the one in our exercise. By applying the natural logarithm, denoted as \(\ln\), we can effectively "undo" the exponential effect of Euler's number, \(e\). In our specific problem, once we isolated the exponential part, we reached the equation: \[ \frac{1}{3} = e^{-0.087t} \] To solve for \(t\), we take the natural logarithm of both sides: \[ \ln\left(\frac{1}{3}\right) = \ln\left(e^{-0.087t}\right) \] The property of logarithms \(\ln(e^x) = x\) lets us simplify the right-hand side, resulting in: \[ \ln\left(\frac{1}{3}\right) = -0.087t \] This simplification makes it much easier to solve for \(t\) by isolating it with simple division, showcasing the natural log's power in handling exponentials.

Decay Rate

The decay rate is a critical concept when dealing with radioactive decay, as it tells us how fast a substance's mass is decreasing over time. In the equation we worked with: \[ m(t) = 15 e^{-0.087t} \] The decay rate is represented by the constant \(-0.087\).

• This rate signifies the percentage of the substance that decreases per time unit, in this case, days.
• The negative sign indicates a reduction over time.

In practical terms, the larger the absolute value of the decay rate, the quicker the decay process occurs. By understanding and calculating this rate, we can predict the time it will take for a substance to reach a certain mass. Knowing the decay rate allows us to plan and make predictions, which are crucial in fields such as medicine, archaeology, and nuclear physics.