Answer :
To determine which choice can be defined using an exponential function, let's look at each option separately:
Choice A:
- On January 1, you receive [tex]$0.10.
- Each subsequent day, the amount doubles: $[/tex]0.20 on January 2, [tex]$0.40 on January 3, $[/tex]0.80 on January 4, and so forth.
This pattern follows exponential growth because the amount is multiplied by a constant factor (2) each day. An exponential function can be used to represent this type of growth. The function for the amount received on day [tex]\( t \)[/tex] can be written as:
[tex]\[ A_t = 0.10 \times (2)^{t-1} \][/tex]
Choice B:
- On January 1, you receive [tex]$5.00.
- Each subsequent day, the amount increases by $[/tex]5.00: [tex]$10.00 on January 2, $[/tex]15.00 on January 3, and so on.
This pattern follows an arithmetic progression because a fixed amount ($5.00) is added each day. An arithmetic sequence can be represented by a linear function. The function for the amount received on day [tex]\( t \)[/tex] can be written as:
[tex]\[ A_t = 5.00 + 5.00(t-1) \][/tex]
Conclusion:
Choice A involves doubling the amount each day, which is characteristic of exponential growth. Therefore, the function for Choice A, [tex]\( A_t = 0.10(2)^{t-1} \)[/tex], represents an exponential function.
This is the choice that can be defined using an exponential function.
Choice A:
- On January 1, you receive [tex]$0.10.
- Each subsequent day, the amount doubles: $[/tex]0.20 on January 2, [tex]$0.40 on January 3, $[/tex]0.80 on January 4, and so forth.
This pattern follows exponential growth because the amount is multiplied by a constant factor (2) each day. An exponential function can be used to represent this type of growth. The function for the amount received on day [tex]\( t \)[/tex] can be written as:
[tex]\[ A_t = 0.10 \times (2)^{t-1} \][/tex]
Choice B:
- On January 1, you receive [tex]$5.00.
- Each subsequent day, the amount increases by $[/tex]5.00: [tex]$10.00 on January 2, $[/tex]15.00 on January 3, and so on.
This pattern follows an arithmetic progression because a fixed amount ($5.00) is added each day. An arithmetic sequence can be represented by a linear function. The function for the amount received on day [tex]\( t \)[/tex] can be written as:
[tex]\[ A_t = 5.00 + 5.00(t-1) \][/tex]
Conclusion:
Choice A involves doubling the amount each day, which is characteristic of exponential growth. Therefore, the function for Choice A, [tex]\( A_t = 0.10(2)^{t-1} \)[/tex], represents an exponential function.
This is the choice that can be defined using an exponential function.
