Answer :
- Choice A represents an exponential growth where the amount doubles each day, starting from $0.10.
- Choice B represents a linear growth where the amount increases by $5.00 each day, starting from $5.00.
- The exponential function for Choice A is given by $A_t = 0.10(2)^{t-1}$.
- Therefore, Choice A can be defined using an exponential function, and the function is $\boxed{A_t = 0.10(2)^{t-1}}$.
### Explanation
1. Understanding the Choices
We are given two choices for a prize. Choice A starts with $0.10$ and doubles each day. Choice B starts with $5.00$ and increases by $5.00$ each day. We need to determine which choice can be represented by an exponential function and find that function.
2. Analyzing Choice A
Choice A: The amount on day $t$ can be represented as $A_t = 0.10 \times 2^{t-1}$. This is because on day 1 ($t=1$), the amount is $0.10 \times 2^{1-1} = 0.10 \times 2^0 = 0.10$. On day 2 ($t=2$), the amount is $0.10 \times 2^{2-1} = 0.10 \times 2^1 = 0.20$. On day 3 ($t=3$), the amount is $0.10 \times 2^{3-1} = 0.10 \times 2^2 = 0.40$, and so on. This is an exponential function.
3. Analyzing Choice B
Choice B: The amount on day $t$ can be represented as $A_t = 5.00 + 5.00(t-1)$. This is because on day 1 ($t=1$), the amount is $5.00 + 5.00(1-1) = 5.00$. On day 2 ($t=2$), the amount is $5.00 + 5.00(2-1) = 10.00$. On day 3 ($t=3$), the amount is $5.00 + 5.00(3-1) = 15.00$, and so on. This is a linear function.
4. Conclusion
Therefore, Choice A can be defined using an exponential function, and the function is $A_t = 0.10(2)^{t-1}$.
### Examples
Exponential functions are used to model various real-world phenomena, such as population growth, compound interest, and radioactive decay. For example, if you invest money in a bank account that earns compound interest, the amount of money in your account will grow exponentially over time. Understanding exponential functions can help you make informed decisions about investments and other financial matters.
- Choice B represents a linear growth where the amount increases by $5.00 each day, starting from $5.00.
- The exponential function for Choice A is given by $A_t = 0.10(2)^{t-1}$.
- Therefore, Choice A can be defined using an exponential function, and the function is $\boxed{A_t = 0.10(2)^{t-1}}$.
### Explanation
1. Understanding the Choices
We are given two choices for a prize. Choice A starts with $0.10$ and doubles each day. Choice B starts with $5.00$ and increases by $5.00$ each day. We need to determine which choice can be represented by an exponential function and find that function.
2. Analyzing Choice A
Choice A: The amount on day $t$ can be represented as $A_t = 0.10 \times 2^{t-1}$. This is because on day 1 ($t=1$), the amount is $0.10 \times 2^{1-1} = 0.10 \times 2^0 = 0.10$. On day 2 ($t=2$), the amount is $0.10 \times 2^{2-1} = 0.10 \times 2^1 = 0.20$. On day 3 ($t=3$), the amount is $0.10 \times 2^{3-1} = 0.10 \times 2^2 = 0.40$, and so on. This is an exponential function.
3. Analyzing Choice B
Choice B: The amount on day $t$ can be represented as $A_t = 5.00 + 5.00(t-1)$. This is because on day 1 ($t=1$), the amount is $5.00 + 5.00(1-1) = 5.00$. On day 2 ($t=2$), the amount is $5.00 + 5.00(2-1) = 10.00$. On day 3 ($t=3$), the amount is $5.00 + 5.00(3-1) = 15.00$, and so on. This is a linear function.
4. Conclusion
Therefore, Choice A can be defined using an exponential function, and the function is $A_t = 0.10(2)^{t-1}$.
### Examples
Exponential functions are used to model various real-world phenomena, such as population growth, compound interest, and radioactive decay. For example, if you invest money in a bank account that earns compound interest, the amount of money in your account will grow exponentially over time. Understanding exponential functions can help you make informed decisions about investments and other financial matters.
