Decay Expression: Value After 10 Hours Calculation

Have you ever wondered how to calculate the remaining value of something that's constantly decreasing, like the depreciation of an asset or the decay of a radioactive substance? This article breaks down the process, especially when dealing with continuous decay. Let's dive into a specific scenario and explore the formula and expression needed to find the value after a certain time.

Understanding Continuous Decay

When we talk about continuous decay, we're referring to a situation where a quantity decreases at a rate that is proportional to its current value. This means the decay happens constantly, not just at specific intervals. A classic example is radioactive decay, where the amount of a radioactive substance decreases continuously over time. But how do we put this into mathematical terms?

The formula for continuous decay is a cornerstone in understanding these phenomena. It allows us to predict the future value of a decaying quantity with precision. This formula is expressed as:

• A(t) = A₀ * e^(kt)

Let's break down what each part of this formula means:

• A(t): This represents the amount or value of the quantity at time 't'. It's what we're trying to find – the value after a certain period has passed.
• A₀: This is the initial amount or value of the quantity. It's the starting point before any decay has occurred. Think of it as the original investment, the initial population, or the starting amount of a radioactive substance.
• e: This is the base of the natural logarithm, an irrational number approximately equal to 2.71828. It's a fundamental constant in mathematics and appears in many contexts, especially when dealing with exponential growth and decay.
• k: This is the continuous decay rate. It's a crucial factor that determines how quickly the quantity decreases. If 'k' is negative, it indicates decay; if it's positive, it indicates growth. The rate is often expressed as a percentage, but in the formula, it needs to be converted to a decimal.
• t: This is the time elapsed, usually measured in hours, years, or any other consistent unit. It's the duration over which the decay occurs.

In essence, this formula tells us that the amount at any time 't' is equal to the initial amount multiplied by the exponential function e raised to the power of the decay rate times time. The negative sign in the exponent (when 'k' is negative) is what makes the function represent decay rather than growth. Understanding each component of this formula is essential for applying it correctly and interpreting the results.

Applying the Formula: A Step-by-Step Guide

Now that we understand the formula, let's see how we can use it to solve practical problems. Here’s a step-by-step guide to applying the continuous decay formula effectively:

1. Identify the Initial Value (A₀): The first step is to determine the starting amount or value of the quantity. This is your A₀. Look for phrases like