To determine which of the provided functions is equivalent to [tex]\( f(x) = 3.02^t \)[/tex], we need to evaluate each candidate function at [tex]\( t = 1 \)[/tex] and compare their results with [tex]\( 3.02^1 = 3.02 \)[/tex]. This process will help us find the function that closely matches the given exponential function.
Let's consider the exponential functions given as options:
1. [tex]\( A. f(x) = 1.202^{5t} \)[/tex]
2. [tex]\( B. f(x) = 1.318^{5t} \)[/tex]
3. [tex]\( C. f(x) = 1.738^{5t} \)[/tex]
4. [tex]\( D. f(x) = 1.247^{5t} \)[/tex]
We will now evaluate each function at [tex]\( t = 1 \)[/tex]:
1. For option A:
[tex]\[
f(1) = 1.202^5
\][/tex]
This value is approximately [tex]\( 1.202^5 = 2.48832 \)[/tex].
2. For option B:
[tex]\[
f(1) = 1.318^5
\][/tex]
This value is approximately [tex]\( 1.318^5 = 3.15349 \)[/tex].
3. For option C:
[tex]\[
f(1) = 1.738^5
\][/tex]
This value is approximately [tex]\( 1.738^5 = 14.44396 \)[/tex].
4. For option D:
[tex]\[
f(1) = 1.247^5
\][/tex]
This value is approximately [tex]\( 1.247^5 = 3.01531 \)[/tex].
Next, we need to compare these values with [tex]\( 3.02 \)[/tex] to find the closest match:
- [tex]\( 2.48832 \)[/tex] (option A) is quite different from [tex]\( 3.02 \)[/tex].
- [tex]\( 3.15349 \)[/tex] (option B) is slightly different from [tex]\( 3.02 \)[/tex].
- [tex]\( 14.44396 \)[/tex] (option C) is significantly different from [tex]\( 3.02 \)[/tex].
- [tex]\( 3.01531 \)[/tex] (option D) is very close to [tex]\( 3.02 \)[/tex].
We observe that the value calculated for option D closely matches [tex]\( 3.02 \)[/tex]. Thus, the function [tex]\( f(x) = 1.247^{5t} \)[/tex] is the best match and hence equivalent to [tex]\( f(x) = 3.02^t \)[/tex].
Therefore, the correct answer is:
D. [tex]\( f(x) = 1.247^{5t} \)[/tex]
