- Analyze the workforce data, which shows a large initial increase that subsequently slows down.
- Evaluate the growth characteristics: Linear models have constant growth, concave up models (quadratic \(a>0\)) have increasing growth, and concave down models (square root) have decreasing growth.
- Match the description of rapid initial growth followed by slowing growth to the concave-down nature of the square root function.
- Conclude that the best model for the data is \(\boxed{\text{a square root function}}\).
Explanation
- Analysis of the Problem Data and Description
The provided data describes the growth of a company's workforce over 10 years, where the number of employees increases from 100 to 980. The goal is to determine the best model for this data: a linear function, a quadratic function with \(a < 0\), a quadratic function with \(a > 0\), or a square root function. The problem specifically states that the company "increased the size of its workforce very quickly at first and then more slowly," which is a hallmark description of a function whose rate of growth is decreasing over time (concave down).
- Examining Growth Rates
We examine the growth rates (first differences) between consecutive years:
- From 1 to 2: \(348 - 100 = 248\)
- From 2 to 3: \(405 - 348 = 57\)
- From 3 to 4: \(575 - 405 = 170\)
- From 4 to 5: \(654 - 575 = 79\)
- From 5 to 6: \(704 - 654 = 50\)
- From 6 to 7: \(746 - 704 = 42\)
- From 7 to 8: \(823 - 746 = 77\)
- From 8 to 9: \(878 - 823 = 55\)
- From 9 to 10: \(980 - 878 = 102\)
While the growth rate fluctuates, the overall trend reflects rapid initial growth followed by a more moderate, slowing rate of increase as the total number of employees grows larger. This shape is characteristic of a concave down function.
- Selecting the Best Model
Evaluating the options:
- A linear function with a positive slope would imply a constant growth rate, which the data clearly does not exhibit.
- A quadratic function with \(a < 0\) is concave down, but it would eventually decrease, which is inconsistent with an increasing number of employees.
- A quadratic function with \(a > 0\) is concave up, meaning the growth rate would be increasing, which contradicts the description that growth slowed down.
- A square root function is concave down (its derivative decreases as \(x\) increases) and remains strictly increasing, perfectly aligning with the description of "quickly at first and then more slowly." Thus, the square root function is the best model.
- Conclusion
Based on the analysis, the square root function is the only model that consistently reflects the rapid early growth followed by a slowing growth rate while maintaining an upward trend throughout the period. Therefore, the correct choice is the square root function. The final answer is \(\boxed{\text{a square root function}}\).
Examples
Consider a real-world scenario where you are filling a cylindrical tank with water. At the start, the volume increases at a steady pace, but as the water level approaches the top, it may be measured differently depending on the input mechanism, or consider a situation like the adoption rate of a new technology: it often starts with a group of early adopters growing rapidly (the "quick start") and then spreads more slowly through the rest of the population (the "slowing growth"). Modeling these trends with functions like square roots helps businesses predict future growth, manage resources, and understand the saturation point of their services.