This is an exponential growth situation. Here's a detailed step-by-step explanation:
1. Understanding the Problem:
Purdue currently has 1320 employees and plans to increase the number of employees by 20% each year. We need to determine whether the growth is linear or exponential, and then select the appropriate equation that describes this growth.
2. Characteristics of Linear Growth:
- Linear growth increases by a fixed amount each period.
- The general form of a linear equation is [tex]\( f(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] is the rate of change (or slope) and [tex]\( b \)[/tex] is the initial value.
3. Characteristics of Exponential Growth:
- Exponential growth increases by a fixed percentage each period.
- The general form of an exponential equation is [tex]\( f(x) = a(1 + r)^x \)[/tex], where [tex]\( a \)[/tex] is the initial value, [tex]\( r \)[/tex] is the growth rate, and [tex]\( x \)[/tex] is the number of periods.
4. Analyzing the Given Growth:
- Purdue’s employee growth is increasing by a fixed percentage (20%) each year. This is a key indicator of exponential growth.
5. Identifying the Correct Exponential Equation:
- The exponential growth formula suitable for this situation is [tex]\( f(x) = 1320(1 + 0.20)^x \)[/tex]. Simplifying, this becomes [tex]\( f(x) = 1320(1.20)^x \)[/tex].
6. Evaluating the Given Options:
- Linear: [tex]\( f(x) = 1320(1.20)^x \)[/tex]: This is incorrectly labeled as linear; it's actually an exponential equation.
- Exponential: [tex]\( f(x) = 1320(0.20)^x \)[/tex]: This form suggests a different growth model, which is incorrect.
- Linear: [tex]\( 1320 + 0.20 x \)[/tex]: This suggests a fixed increase, which does not apply here.
- Exponential: [tex]\( 1320 + 1.20 x \)[/tex]: This incorrectly combines linear and exponential components.
Given these points, the correct equation describing this exponential growth situation is:
[tex]\[ f(x) = 1320(1.20)^x \][/tex]
This describes an exponential growth model where the number of employees increases by 20% each year.
