To analyze the given exponential function [tex]\( v(t) = 27,500 (1.12)^t \)[/tex], let's break down the components and understand what each part represents.
1. Initial value of the car:
- The initial value of an exponential function [tex]\( v(t) = a \cdot b^t \)[/tex] is given by the coefficient [tex]\( a \)[/tex] when [tex]\( t = 0 \)[/tex].
- Here, the function is [tex]\( v(t) = 27,500 (1.12)^t \)[/tex].
- The coefficient [tex]\( a \)[/tex] is 27,500.
Therefore, the initial value of the car is:
[tex]\[
\boxed{27,500}
\][/tex]
2. Growth or decay:
- To determine whether the function represents growth or decay, we look at the base [tex]\( b \)[/tex] of the exponential function [tex]\( a \cdot b^t \)[/tex].
- If [tex]\( b > 1 \)[/tex], the function represents growth.
- If [tex]\( 0 < b < 1 \)[/tex], the function represents decay.
- In the given function, [tex]\( b \)[/tex] is 1.12.
Since 1.12 is greater than 1, the function represents:
[tex]\[
\boxed{\text{growth}}
\][/tex]
3. Percent change each year:
- The base of the exponential function indicates the growth factor.
- We can find the annual percent change by subtracting 1 from the growth factor and then converting it to a percentage.
- Here, the growth factor is 1.12.
- The percent change is [tex]\((1.12 - 1) \times 100\%\)[/tex].
Therefore, the value of the car changes each year by:
[tex]\[
\boxed{12.00\%}
\][/tex]
To summarize:
1. Initial value of the car: [tex]\( \boxed{27,500} \)[/tex]
2. Represents: [tex]\( \boxed{\text{growth}} \)[/tex]
3. Percent change each year: [tex]\( \boxed{12.00\%} \)[/tex]
