Problem 31 - Algebra 1

Question

Use the following information. From 1894 to 1903 the number of miles of cable car track decreased by about \(10 \%\) per year. There were 302 miles of track in 1894 . Write an exponential decay model showing the number of miles \(M\) of cable car track left in year \(t.\)

Step-by-Step Solution

Answer

The exponential decay model showing the number of miles \(M\) of cable car track left in year \(t\) is expressed by the equation \(M(t) = 302 * (0.90)^t\).

Step 1: Understand the Exercise

The exercise provides that in 1894 there were 302 miles of track and this decreased by about 10 \% per year. This indicates that the amount of track is reducing in an exponential way. Therefore, an exponential decay model should be created.

Step 2: Identify the Decay Rate and Initial Value

The decay rate, \(r\), is the rate at which the number of miles is decreasing each year, in this case, 10\% or 0.10. The initial value, \(M_0\), is the number of miles in 1894, which we know to be 302 miles.

Step 3: Writing the Exponential Decay Equation

Substituting the values of \(M_0\) and \(r\) into the equation for exponential decay, we get the equation \(M(t) = 302 * (1 - 0.10)^t\). This is the model showing the number of miles \(M\) of cable car track left in year \(t\).

Key Concepts

Decay Rate

In the context of exponential decay, the decay rate is an essential concept that defines how fast a quantity decreases over time. When a quantity decreases by a fixed percentage each year, such as 10% in this case, we call this percentage the "decay rate." Let's break it down further:

• The decay rate is represented as a decimal in mathematical equations. So, 10% becomes 0.10.
• This rate tells us that each year, the quantity will be 90% of what it was the previous year.

Plugging the decay rate into equations is crucial for accurate predictions. In practical situations, understanding this rate helps determine how quickly resources are depleting, such as the miles of cable car track dwindling over time.
By understanding and calculating the decay rate, we can predict future values accurately using mathematical models.

Initial Value

The initial value in an exponential decay equation is the starting point of the quantity that is subject to decay. In our exercise, the initial value is the number of miles of cable car track available at the beginning of the observation period, which is 302 miles in 1894.Here's why the initial value is important:

• The initial value provides a reference point for any further calculations. It is the base from which decay is measured.
• It directly influences the absolute amount of the quantity over time. Larger initial values result in larger values throughout the decay process.

Understanding the initial value is crucial in many real-life applications, such as budgeting, inventory management, and assessing historical data trends. In mathematical terms, this value is typically denoted as \( M_0 \) in decay equations, serving as a vital component in forming the entire model.

Exponential Equation

An exponential decay equation represents how a quantity decreases exponentially over time. It is a mathematical expression that models systems or situations where the rate of change is proportional to the current value, common in natural processes and even in technology, like our cable car track example.The standard form of an exponential decay equation is: \[ M(t) = M_0 \times (1 - r)^t \]Where:

• \( M(t) \) is the quantity remaining after time \( t \).
• \( M_0 \) is the initial quantity, the starting point for our calculations.
• \( r \) is the decay rate, the proportion by which the quantity decreases annually.
• \( t \) is the time that has elapsed, often measured in years.

Using this equation, you can replace the variables with specific values from your problem to predict future outcomes. For example, plugging in the initial value of 302 miles, and a decay rate of 0.10 for our cable car context, yields a clear and understandable model to see how miles diminish over the years. This type of equation is powerful in forecasting and making informed decisions based on expected future scenarios.