Solve: Y = 1000 * 0.95^x For X = 0, 1, 2

by Alex Johnson 41 views

Let's dive into solving exponential equations! In this article, we're tackling the equation y = 1000 * 0.95^x. Our goal is to find the value of 'y' when 'x' takes on the values of 0, 1, and 2. This is a classic example of how exponential functions behave, and understanding this will help you in various mathematical and real-world scenarios. So, let's break it down step by step.

Understanding Exponential Equations

Before we jump into the calculations, it’s essential to grasp what an exponential equation is all about. An exponential equation is one where the variable appears in the exponent. In our case, 'x' is the exponent. These equations are used to model various phenomena, such as population growth, radioactive decay, and compound interest. The general form of an exponential equation is y = a * b^x, where 'a' is the initial value, 'b' is the base (the growth or decay factor), and 'x' is the exponent (time or the number of periods). In our specific equation, y = 1000 * 0.95^x, '1000' is the initial value, '0.95' is the decay factor (since it's less than 1), and 'x' is the variable we'll be changing. When dealing with exponential equations, it's crucial to remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This order ensures that we calculate the exponent before multiplying by the coefficient. Exponential functions can either increase rapidly (exponential growth) if the base is greater than 1, or decrease rapidly (exponential decay) if the base is between 0 and 1. Understanding this behavior is key to interpreting the results we'll obtain. For instance, a decay factor of 0.95 suggests that the value of 'y' will decrease as 'x' increases. This is because we're multiplying the initial value by a number less than 1 raised to a power, which will result in progressively smaller values. Exponential equations are powerful tools in many fields, including finance, biology, and physics. Mastering them opens the door to understanding and predicting various real-world phenomena.

Solving for x = 0

Let's start with the simplest case: finding the value of 'y' when 'x' is 0. This is a fundamental concept in exponential functions, and it's crucial to understand how any number raised to the power of 0 behaves. Our equation is y = 1000 * 0.95^x, and we're substituting 'x' with 0. So, we have y = 1000 * 0.95^0. Now, here's the key rule to remember: any non-zero number raised to the power of 0 is 1. This is a mathematical principle that simplifies calculations significantly. Therefore, 0. 95^0 equals 1. Our equation now becomes y = 1000 * 1. This simplifies to y = 1000. So, when 'x' is 0, 'y' is 1000. This result makes sense in the context of exponential functions. When 'x' is 0, it represents the initial value or the starting point. In our equation, '1000' is indeed the initial value, as it's the value of 'y' before any decay (or growth) occurs. This principle is widely used in various applications. For example, in finance, if we're modeling the depreciation of an asset, the value of the asset at time 0 is its initial cost. In biology, if we're modeling population growth, the population size at time 0 is the initial population. Understanding this concept is essential for interpreting the behavior of exponential functions and their applications in the real world. It provides a baseline or a reference point for understanding how the value changes as 'x' varies. This simple calculation highlights the importance of understanding fundamental mathematical rules, as they often lead to significant simplifications and insights.

Solving for x = 1

Next up, let's find the value of 'y' when 'x' is 1. This step will show us how the value of 'y' changes after the first period or unit of time. Again, we start with our equation: y = 1000 * 0.95^x. This time, we're substituting 'x' with 1, so we have y = 1000 * 0.95^1. Any number raised to the power of 1 is simply the number itself. Therefore, 0. 95^1 equals 0.95. Our equation now becomes y = 1000 * 0.95. To find 'y', we need to multiply 1000 by 0.95. This calculation is straightforward: 1000 * 0.95 = 950. So, when 'x' is 1, 'y' is 950. This result tells us that after one period, the value of 'y' has decreased from 1000 to 950. This decrease is due to the decay factor of 0.95. Since 0.95 is less than 1, it indicates a decay or reduction in value. In real-world terms, this could represent a depreciation of an asset, a decrease in population size, or the decay of a radioactive substance. The amount of the decrease is 1000 - 950 = 50. This means that 'y' has decreased by 50 units after one period. Understanding how the value changes from one period to the next is crucial in analyzing exponential functions. It allows us to predict the behavior of the function over time and to make informed decisions based on the results. In this case, we see that 'y' decreases by 5% (since 0.95 represents 95% of the previous value) in each period. This constant percentage decrease is a characteristic of exponential decay. By calculating the value of 'y' for 'x' equals 1, we gain insight into the rate and direction of change in the function.

Solving for x = 2

Now, let's tackle the case where 'x' is 2. This will give us further insight into the behavior of the exponential equation over time. We start with our familiar equation: y = 1000 * 0.95^x. Substituting 'x' with 2, we get y = 1000 * 0.95^2. This means we need to calculate 0. 95 squared (0.95 * 0.95) first. 0. 95 * 0.95 equals 0.9025. So, our equation becomes y = 1000 * 0.9025. Next, we multiply 1000 by 0.9025. This gives us y = 902.5. Therefore, when 'x' is 2, 'y' is 902.5. This result shows that the value of 'y' continues to decrease as 'x' increases. After two periods, 'y' has decreased from 950 (when x=1) to 902.5. The decrease from the initial value of 1000 is even more significant: 1000 - 902.5 = 97.5. This further emphasizes the concept of exponential decay. The value decreases by a constant percentage in each period, but the absolute amount of the decrease becomes smaller as 'y' gets closer to zero. This is a characteristic feature of exponential decay, where the rate of decay slows down over time. Understanding the value of 'y' when 'x' is 2 helps us to see the trend in the function's behavior. By calculating 'y' for different values of 'x', we can create a clearer picture of how the function changes over time. This is particularly useful in real-world applications where we need to predict future values based on current trends. For example, in finance, we might use this to model the depreciation of an asset over several years. In biology, we could use it to model the decline in a population due to a certain factor.

Summary of Results

Let's recap our findings. We started with the equation y = 1000 * 0.95^x and solved for 'y' at three different values of 'x':

  • When x = 0, we found that y = 1000. This is the initial value of 'y' before any decay occurs.
  • When x = 1, we found that y = 950. This shows the value of 'y' after one period, with a decrease of 50 units from the initial value.
  • When x = 2, we found that y = 902.5. This further illustrates the decay, with 'y' decreasing by another 47.5 units from the previous period.

These results demonstrate the concept of exponential decay. As 'x' increases, 'y' decreases, but the rate of decrease slows down over time. This is because we're multiplying by a decay factor (0.95) that is less than 1. Each time we increase 'x' by 1, we're multiplying the previous value of 'y' by 0.95, resulting in a smaller decrease than the previous period. Understanding this pattern is crucial for interpreting exponential decay in various contexts. For example, in finance, this could represent the depreciation of an asset, where the asset loses value over time, but the amount of the loss decreases each year. In medicine, this could represent the decay of a drug in the bloodstream, where the concentration of the drug decreases over time, but the rate of decrease slows down. The ability to calculate and interpret exponential decay is a valuable skill in many fields. It allows us to make predictions about future values and to understand the long-term behavior of systems that exhibit exponential decay.

Conclusion

In this article, we've successfully solved for 'y' in the equation y = 1000 * 0.95^x for 'x' values of 0, 1, and 2. We've seen how the value of 'y' changes as 'x' increases, demonstrating the concept of exponential decay. Understanding exponential equations is crucial for various applications in mathematics and real-world scenarios. By mastering the techniques discussed here, you'll be well-equipped to tackle more complex problems involving exponential functions. For further exploration on exponential functions, you can visit resources like Khan Academy's Exponential Functions section. Keep practicing, and you'll become more confident in your ability to solve these types of equations!