Answer :
To determine which choice can be defined using an exponential function, let's examine both options provided:
Choice A: Starts with [tex]$0.10 on January 1, and the amount doubles each day. So, on January 2 it's $[/tex]0.20, on January 3 it's [tex]$0.40, and so on. The amounts are 0.10, 0.20, 0.40, 0.80, etc.
This represents exponential growth because the amount is multiplied by 2 each day. We can define an exponential function for this pattern as \( A_t = 0.10 \times 2^{t-1} \), where \( t \) is the day number.
Choice B: Begins with $[/tex]5.00 on the first day, then increases by [tex]$5.00 each subsequent day. So, on the first day, you get $[/tex]5.00, on the second day [tex]$10.00, on the third day $[/tex]15.00, and so forth. The amounts are 5.00, 10.00, 15.00, 20.00, etc.
This pattern shows a linear growth because the amount increases by a fixed value each day. The function for this linear pattern can be expressed as [tex]\( A_t = 5.00 + 5.00(t-1) \)[/tex], where [tex]\( t \)[/tex] is the day number.
Since an exponential function is characterized by a constant multiplicative rate of change (doubling, in this case), Choice A can be defined using an exponential function. The function for Choice A is:
[tex]\[ A_t = 0.10 \times 2^{t-1} \][/tex]
This means on any day [tex]\( t \)[/tex], the amount you receive is 0.10 multiplied by 2 raised to the power of [tex]\( t-1 \)[/tex].
Choice A: Starts with [tex]$0.10 on January 1, and the amount doubles each day. So, on January 2 it's $[/tex]0.20, on January 3 it's [tex]$0.40, and so on. The amounts are 0.10, 0.20, 0.40, 0.80, etc.
This represents exponential growth because the amount is multiplied by 2 each day. We can define an exponential function for this pattern as \( A_t = 0.10 \times 2^{t-1} \), where \( t \) is the day number.
Choice B: Begins with $[/tex]5.00 on the first day, then increases by [tex]$5.00 each subsequent day. So, on the first day, you get $[/tex]5.00, on the second day [tex]$10.00, on the third day $[/tex]15.00, and so forth. The amounts are 5.00, 10.00, 15.00, 20.00, etc.
This pattern shows a linear growth because the amount increases by a fixed value each day. The function for this linear pattern can be expressed as [tex]\( A_t = 5.00 + 5.00(t-1) \)[/tex], where [tex]\( t \)[/tex] is the day number.
Since an exponential function is characterized by a constant multiplicative rate of change (doubling, in this case), Choice A can be defined using an exponential function. The function for Choice A is:
[tex]\[ A_t = 0.10 \times 2^{t-1} \][/tex]
This means on any day [tex]\( t \)[/tex], the amount you receive is 0.10 multiplied by 2 raised to the power of [tex]\( t-1 \)[/tex].
