Answer :
To determine how the number of employees, represented by the function [tex]\( f(t) = 1.5(0.90)^t \)[/tex], is changing over time, let's break down the components of the function.
The given function is an exponential decay function because the base of the exponential, [tex]\( 0.90 \)[/tex], is less than 1. In general, an exponential function of the form [tex]\( f(t) = a \cdot b^t \)[/tex] will represent:
- Exponential decay if [tex]\( 0 < b < 1 \)[/tex]
- Exponential growth if [tex]\( b > 1 \)[/tex]
Here, the initial value [tex]\( a \)[/tex] is 1.5, which means at [tex]\( t = 0 \)[/tex], [tex]\( f(0) = 1.5 \)[/tex]. This represents the initial number of employees in thousands.
The base [tex]\( b = 0.90 \)[/tex] indicates the factor by which the number of employees is multiplied each year. When the base is between 0 and 1, it means the quantity is decreasing.
To find the rate of decrease:
- We know [tex]\( b = 1 - \text{decrease rate} \)[/tex]
- Given [tex]\( b = 0.90 \)[/tex], we can deduce [tex]\( 1 - \text{decrease rate} = 0.90 \)[/tex]
- Subtracting 0.90 from 1, we get [tex]\( \text{decrease rate} = 1 - 0.90 = 0.10 = 10\% \)[/tex]
Therefore, the function [tex]\( f(t) = 1.5(0.90)^t \)[/tex] represents a situation where the number of employees is decreasing by 10% each year.
So, the correct answer is:
B. The number of employees is decreasing by 10% every year.
The given function is an exponential decay function because the base of the exponential, [tex]\( 0.90 \)[/tex], is less than 1. In general, an exponential function of the form [tex]\( f(t) = a \cdot b^t \)[/tex] will represent:
- Exponential decay if [tex]\( 0 < b < 1 \)[/tex]
- Exponential growth if [tex]\( b > 1 \)[/tex]
Here, the initial value [tex]\( a \)[/tex] is 1.5, which means at [tex]\( t = 0 \)[/tex], [tex]\( f(0) = 1.5 \)[/tex]. This represents the initial number of employees in thousands.
The base [tex]\( b = 0.90 \)[/tex] indicates the factor by which the number of employees is multiplied each year. When the base is between 0 and 1, it means the quantity is decreasing.
To find the rate of decrease:
- We know [tex]\( b = 1 - \text{decrease rate} \)[/tex]
- Given [tex]\( b = 0.90 \)[/tex], we can deduce [tex]\( 1 - \text{decrease rate} = 0.90 \)[/tex]
- Subtracting 0.90 from 1, we get [tex]\( \text{decrease rate} = 1 - 0.90 = 0.10 = 10\% \)[/tex]
Therefore, the function [tex]\( f(t) = 1.5(0.90)^t \)[/tex] represents a situation where the number of employees is decreasing by 10% each year.
So, the correct answer is:
B. The number of employees is decreasing by 10% every year.
