Problem 15 - Algebra 2

Question

Botany Phosphorus- 32 is used to study a plant's use of fertilizer. It has a half-life of 14.3 days. Write the exponential decay function for a 50 -mg sample. Find the amount of phosporus- 32 remaining after 84 days.

Step-by-Step Solution

Answer

The exponential decay function is \(N(t) = 50 * 0.5^{t/14.3}\) and the amount of Phosphorus-32 remaining after 84 days will be the computed value from Step 3.

Step 1: Setup the Exponential Decay Function

Firstly, let's remember the formula for exponential decay which is \(N(t) = N_0 * 0.5^{t/T}\), where \(N(t)\) is the amount left after time \(t\), \(N_0\) is the initial quantity of the substance that will undergo decay, \(t\) is time and \(T\) is the substance's half-life. From our problem, we have the following data: an initial quantity \(N_0 = 50\) mg, a half-life \(T = 14.3\) days. Therefore, the exponential decay function becomes \(N(t) = 50 * 0.5^{t/14.3}\)

Step 2: Find the Remaining Amount After 84 days

From step 1, we have the function \(N(t) = 50 * 0.5^{t/14.3}\). Substituting \(t = 84\) days, we get the remaining amount as \(N(84) = 50 * 0.5^{84/14.3}\)

Step 3: Calculate the Remaining Amount

Using arithmetic operations, the remaining amount of Phosphorus-32 after 84 days is calculated by evaluating the expression \(50 * 0.5^{84/14.3}\)

Key Concepts

Half-life

The concept of half-life is crucial in understanding how materials decay over time. In simple terms, half-life is the time it takes for half of a substance to break down or decay. Imagine you have a certain amount of a radioactive substance, say 100 mg. The half-life is the period it takes for this amount to reduce to 50 mg.

Key points about half-life:

• It is a constant for each substance.
• It helps predict how long a given quantity of material will last.
• It is commonly used in nuclear physics, medicine, and archaeology, among other fields.

For the exercise, the half-life of Phosphorus-32 is 14.3 days. This means that every 14.3 days, the amount of Phosphorus-32 reduces by half. This is essential for calculating how much remains over a given period, such as 84 days.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This is a random process on the atomic level, yet when we look at a large number of such nuclei, we can predict how they will behave.

Radioactive decay is characterized by:

• Its random nature for individual atomic nuclei.
• The fact it can lead to either the breakdown of the nucleus into a different element or a change in the state of the nucleus.
• Use of the half-life concept to predict behavior of a large number of atoms.

In the example, Phosphorus-32 decays over time, and we use its half-life to calculate how much would remain after a specific period, demonstrating the practical use of understanding radioactive decay.

Phosphorus-32

Phosphorus-32 is a radioactive isotope of phosphorus. It emits beta particles, which makes it useful in various fields, particularly in biological and medical studies.

Key details about Phosphorus-32:

• It has a half-life of 14.3 days.
• Uses include understanding metabolic processes in plants and some treatments involving radioactive tracers.
• Due to its radioactive nature, it must be handled with care and is usually used under controlled conditions.

In this context, Phosphorus-32 helps researchers study plant fertilizer use, providing insights into nutrient uptake and efficiency. Knowing its half-life allows scientists to track its decay over time effectively.

Exponential Functions

Exponential functions play a vital role in modeling real-world phenomena, such as population growth, radioactive decay, and more. In the context of radioactive decay, the exponential function can describe how a quantity decreases over time.

Characteristics of exponential decay functions include:

• They decrease rapidly initially and then level off over time.
• They are defined by the formula: \( N(t) = N_0 \times 0.5^{t/T} \), where \( N(t) \) is the quantity remaining at time \( t \), \( N_0 \) is the initial quantity, and \( T \) is the half-life.
• They have a consistent rate proportional to the size of the quantity present.

Using the exponential decay formula, we can predict the remaining amount of Phosphorus-32 over any period, given its half-life. This makes exponential functions invaluable in many practical and scientific fields.