Problem 15 - Algebra A Combined Function

Question

Solve. Unless noted otherwise, round answers to the nearest whole. The number of employees for a certain company has been decreasing each year by \(5 \%\). If the company currently has 640 employees and this rate continues, find the number of employees in 10 years.

Step-by-Step Solution

Answer

383 employees.

Step 1: Understand the Problem

We need to find the number of employees the company will have in 10 years given that the number decreases by 5% each year from the current number of 640 employees.

Step 2: Identify the Exponential Decay Formula

The number of employees can be modeled by an exponential decay formula: \[ N = N_0 \times (1 - r)^t \]where:- \(N\) is the final amount after time \(t\),- \(N_0\) is the initial amount (640 employees),- \(r\) is the decay rate (5%, or 0.05), and- \(t\) is the time in years (10 years).

Step 3: Substitute Values into the Formula

Substitute the given values into the formula:\[ N = 640 \times (1 - 0.05)^{10} \]This simplifies to:\[ N = 640 \times (0.95)^{10} \]

Step 4: Calculate the Exponential Decay

Calculate \((0.95)^{10}\):\[(0.95)^{10} \approx 0.59874\]Now substitute this back into the equation:\[ N = 640 \times 0.59874 \]

Step 5: Compute the Final Number of Employees

Multiply 640 by 0.59874 to find the projected number of employees:\[ N \approx 640 \times 0.59874 = 383.1936 \]

Step 6: Round to the Nearest Whole Number

Round 383.1936 to the nearest whole number: The approximate number of employees in 10 years is 383.

Key Concepts

Exponential Decay Formula

In mathematical terms, exponential decay describes a process where a quantity decreases at a constant rate over time. To model this kind of decrease, we use the exponential decay formula:\[ N = N_0 \times (1 - r)^t \]- **\(N\)** represents the future value of the quantity, after time \(t\).- **\(N_0\)** is the initial value before the decay starts.- **\(r\)** signifies the rate of decrease, often expressed as a decimal.- **\(t\)** is the time period over which the decay occurs.This formula is particularly useful for predicting the future value of a variable that decreases over time, such as the number of employees in a company. Understanding this formula allows you to determine how much a specific quantity will diminish over a given period. In our exercise, we're calculating how a workforce of 640 employees will shrink by 5% annually over 10 years.

Decay Rate

The decay rate is a critical component in an exponential decay problem because it determines how quickly the quantity diminishes. In the exercise, the decay rate was a constant 5% per year. This means each year's workforce reduces to 95% of its previous size. Here's how you set it up:- **Convert the percentage to a decimal:** - 5% becomes 0.05 - Make sure to subtract from 1 to show reduction: \[1 - 0.05 = 0.95\]The decay rate plays a significant role in calculations as it affects the speed at which the original quantity depletes over time. Using decay rates accurately ensures your predictions are realistic and reflect the true nature of the reduction process.

Rounding Numbers

Rounding numbers is an important step in many calculations, especially when dealing with real-world data that requires precision but also clarity. In our exercise, after calculating the projected number of employees after 10 years, we reached an answer of roughly 383.1936. Why and how do we round this? - **Why round?** - Easier interpretation: Understanding an exact number like 383 is simpler than 383.1936. - Most contexts, like employment numbers, naturally use whole numbers. - **How to round:** - Look at the number immediately after the decimal. - If it is 5 or greater, round up. - If it is less than 5, round down. In this case, we rounded 383.1936 to 383, because the digit after the decimal is "1", which is less than 5. Thus, rounding is not just about simplifying but also enhancing the practicality and communication of your results.