Let's walk through the problem step-by-step to interpret the function [tex]\( a(t) = 500 \times 1.05^t \)[/tex] and understand its context within the problem of a savings account.
1. Identify the Initial Deposit:
- The function given is [tex]\( a(t) = 500 \times 1.05^t \)[/tex].
- Here, the number "500" represents the initial deposit amount. This means the savings account was opened with an initial deposit of $500.
2. Understand the Growth Factor:
- The term [tex]\( 1.05 \)[/tex] in the function [tex]\( 500 \times 1.05^t \)[/tex] is called the growth factor.
- A growth factor of 1.05 indicates that the value of the account grows by a rate of 5% per time period (typically a year).
3. Determine the Annual Growth Rate:
- The growth factor of 1.05 can be broken down as follows: 1 (representing 100% of the original amount) + 0.05 (representing an additional 5% growth).
4. Interpret the Percent Rate of Change:
- The correct interpretation of the rate of change must focus on the context of increasing or decreasing values.
- Since [tex]\( 1.05 \)[/tex] means an increase, not a decrease, we can safely exclude any options involving decreases.
5. Select the Correct Interpretation:
- We now have two viable interpretations left:
- The percent rate of change is 5%, which means the value of the account increases by 5% per year.
- The percent rate of change is 95%, which means the value of the account increases by 5% per year.
- Comparing these two, only the correct option aligns with both the provided growth factor and the description of increase.
Therefore, the correct interpretation in the context of the given problem is:
- The percent rate of change is 5%, which means the value of the account increases by 5% per year.
And hence, the correct choice is:
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- The percent rate of change is 5%, which means the value of the account increases by 5% per year.
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