Select the correct answer.

The following function describes the number of employees working at a company, in thousands, where [tex] t [/tex] represents the number of years since the company revised the benefits package.
[tex]\[ f(t) = 1.5(0.90)^t \][/tex]

A. The number of employees is decreasing by 90% every year.
B. The number of employees is decreasing by 10% every year.
C. The number of employees is increasing by 50% every year.
D. The number of employees is increasing by 90% every year.



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.