Answer :
To determine which exponential function best represents the relationship between [tex]\( f(t) \)[/tex] and [tex]\( t \)[/tex], let's go through each given function and check whether it fits the provided data points.
The data points given are:
- At [tex]\( t = 0 \)[/tex], [tex]\( f(0) = 180 \)[/tex]
- At [tex]\( t = 1 \)[/tex], [tex]\( f(1) = 90 \)[/tex]
- At [tex]\( t = 2 \)[/tex], [tex]\( f(2) = 45 \)[/tex]
### Checking [tex]\( f(t) = 0.5(180)^t \)[/tex]
1. Calculate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 0.5 \times (180)^0 = 0.5 \times 1 = 0.5 \][/tex]
This does not match [tex]\( f(0) = 180 \)[/tex].
Since [tex]\( f(0) \)[/tex] does not match, [tex]\( f(t) = 0.5(180)^t \)[/tex] is not the correct function.
### Checking [tex]\( f(t) = 180(0.25)^t \)[/tex]
1. Calculate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 180 \times (0.25)^0 = 180 \times 1 = 180 \][/tex]
This matches [tex]\( f(0) = 180 \)[/tex].
2. Calculate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 180 \times (0.25)^1 = 180 \times 0.25 = 45 \][/tex]
This does not match [tex]\( f(1) = 90 \)[/tex].
Since [tex]\( f(1) \)[/tex] does not match, [tex]\( f(t) = 180(0.25)^t \)[/tex] is not the correct function.
### Checking [tex]\( f(t) = 180(0.5)^t \)[/tex]
1. Calculate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 180 \times (0.5)^0 = 180 \times 1 = 180 \][/tex]
This matches [tex]\( f(0) = 180 \)[/tex].
2. Calculate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 180 \times (0.5)^1 = 180 \times 0.5 = 90 \][/tex]
This matches [tex]\( f(1) = 90 \)[/tex].
3. Calculate [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = 180 \times (0.5)^2 = 180 \times 0.25 = 45 \][/tex]
This matches [tex]\( f(2) = 45 \)[/tex].
Since all calculated [tex]\( f(t) \)[/tex] values match the given data points, [tex]\( f(t) = 180(0.5)^t \)[/tex] is the correct function.
### Checking [tex]\( f(t) = 0.5(50)^t \)[/tex]
1. Calculate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 0.5 \times (50)^0 = 0.5 \times 1 = 0.5 \][/tex]
This does not match [tex]\( f(0) = 180 \)[/tex].
Since [tex]\( f(0) \)[/tex] does not match, [tex]\( f(t) = 0.5(50)^t \)[/tex] is not the correct function.
### Conclusion
The exponential function that best represents the relationship between [tex]\( f(t) \)[/tex] and [tex]\( t \)[/tex] is:
[tex]\[ f(t) = 180(0.5)^t \][/tex]
The data points given are:
- At [tex]\( t = 0 \)[/tex], [tex]\( f(0) = 180 \)[/tex]
- At [tex]\( t = 1 \)[/tex], [tex]\( f(1) = 90 \)[/tex]
- At [tex]\( t = 2 \)[/tex], [tex]\( f(2) = 45 \)[/tex]
### Checking [tex]\( f(t) = 0.5(180)^t \)[/tex]
1. Calculate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 0.5 \times (180)^0 = 0.5 \times 1 = 0.5 \][/tex]
This does not match [tex]\( f(0) = 180 \)[/tex].
Since [tex]\( f(0) \)[/tex] does not match, [tex]\( f(t) = 0.5(180)^t \)[/tex] is not the correct function.
### Checking [tex]\( f(t) = 180(0.25)^t \)[/tex]
1. Calculate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 180 \times (0.25)^0 = 180 \times 1 = 180 \][/tex]
This matches [tex]\( f(0) = 180 \)[/tex].
2. Calculate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 180 \times (0.25)^1 = 180 \times 0.25 = 45 \][/tex]
This does not match [tex]\( f(1) = 90 \)[/tex].
Since [tex]\( f(1) \)[/tex] does not match, [tex]\( f(t) = 180(0.25)^t \)[/tex] is not the correct function.
### Checking [tex]\( f(t) = 180(0.5)^t \)[/tex]
1. Calculate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 180 \times (0.5)^0 = 180 \times 1 = 180 \][/tex]
This matches [tex]\( f(0) = 180 \)[/tex].
2. Calculate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 180 \times (0.5)^1 = 180 \times 0.5 = 90 \][/tex]
This matches [tex]\( f(1) = 90 \)[/tex].
3. Calculate [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = 180 \times (0.5)^2 = 180 \times 0.25 = 45 \][/tex]
This matches [tex]\( f(2) = 45 \)[/tex].
Since all calculated [tex]\( f(t) \)[/tex] values match the given data points, [tex]\( f(t) = 180(0.5)^t \)[/tex] is the correct function.
### Checking [tex]\( f(t) = 0.5(50)^t \)[/tex]
1. Calculate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 0.5 \times (50)^0 = 0.5 \times 1 = 0.5 \][/tex]
This does not match [tex]\( f(0) = 180 \)[/tex].
Since [tex]\( f(0) \)[/tex] does not match, [tex]\( f(t) = 0.5(50)^t \)[/tex] is not the correct function.
### Conclusion
The exponential function that best represents the relationship between [tex]\( f(t) \)[/tex] and [tex]\( t \)[/tex] is:
[tex]\[ f(t) = 180(0.5)^t \][/tex]
