To determine how the function [tex]\( g(t) = 5 \cdot 1.45^t \)[/tex] changes over the interval from [tex]\( t = 4 \)[/tex] to [tex]\( t = 5 \)[/tex], we will follow these steps:
1. Calculate [tex]\( g(4) \)[/tex]:
[tex]\[
g(4) = 5 \cdot 1.45^4
\][/tex]
Given the result, [tex]\( g(4) \approx 22.10253125 \)[/tex].
2. Calculate [tex]\( g(5) \)[/tex]:
[tex]\[
g(5) = 5 \cdot 1.45^5
\][/tex]
Given the result, [tex]\( g(5) \approx 32.0486703125 \)[/tex].
3. Determine the ratio of [tex]\( g(t) \)[/tex] at [tex]\( t = 5 \)[/tex] to [tex]\( g(t) \)[/tex] at [tex]\( t = 4 \)[/tex]:
[tex]\[
\text{Ratio} = \frac{g(5)}{g(4)} = \frac{32.04867031249999}{22.10253125} \approx 1.45
\][/tex]
4. Convert this ratio to a percentage increase. Since the ratio is approximately [tex]\( 1.45 \)[/tex], we calculate the percentage increase as follows:
[tex]\[
\text{Percentage Increase} = (1.45 - 1) \times 100 = 0.45 \times 100 = 45\%
\][/tex]
5. Interpret the result:
- The function [tex]\( g(t) \)[/tex] increases by a factor of [tex]\( 1.45 \)[/tex] from [tex]\( t = 4 \)[/tex] to [tex]\( t = 5 \)[/tex].
- Alternatively, we can say that [tex]\( g(t) \)[/tex] increases by [tex]\( 45\% \)[/tex] over this interval.
- Note that [tex]\( 145\% \)[/tex] represents a much greater increase and is not the correct interpretation.
Consequently, the correct interpretation for the change in [tex]\( g(t) \)[/tex] over the interval from [tex]\( t = 4 \)[/tex] to [tex]\( t = 5 \)[/tex] is:
[tex]\[
\boxed{\text{g(t) increases by a factor of 1.45}}
\][/tex]
