Select the function that represents Usain Bolt's account balance if he makes one deposit of J$3,000 at Kingston Federal Bank.
Understand the Problem
The question is asking to select the correct function that represents Usain Bolt's account balance after making a deposit of J$3,000 at Kingston Federal Bank. The options provided involve exponential growth equations that relate to how the account balance increases over time.
Answer
$$ J(t) = 3000(1 + 0.05)^t $$
Answer for screen readers
$$ J(t) = 3000(1 + 0.05)^t $$
Steps to Solve
- Understanding the growth model
The account balance grows according to a compound growth function. The general form of the compound interest formula is:
$$ J(t) = P(1 + r)^t $$
where
- ( J(t) ) is the amount in the account at time ( t ),
- ( P ) is the principal amount (initial deposit),
- ( r ) is the growth rate per time period, and
- ( t ) is the number of time periods.
In this case, the initial deposit ( P ) is J$3,000.
- Identifying the correct growth rate
To analyze the provided data, we need to determine the growth rate from the account balances at different times:
- From J$3,000 to J$3,150 at ( t = 1 ):
- Growth = J$3,150 - J$3,000 = J$150
- Growth rate: ( \frac{150}{3000} = 0.05 ) (or 5%)
This indicates that the ( r ) in our model is 0.05.
- Formulating the balance function
Using the growth rate of 0.05 in the compound interest formula, we get:
$$ J(t) = 3000(1 + 0.05)^t $$
This matches one of the options provided in the question.
- Examining the other options
- ( J(t) = 3000(1 - 0.05)^t ) represents a decrease.
- ( J(t) = 3000(1 - 0.95)^t ) doesn't make sense in this context.
- ( J(t) = 3000(1 + 0.005)^t ) suggests a much lower growth rate (0.5%), which does not align with the data.
$$ J(t) = 3000(1 + 0.05)^t $$
More Information
The function ( J(t) = 3000(1 + 0.05)^t ) represents continuous growth at a rate of 5% annually. This is a standard way to express compound interest and accurately reflects the given cumulative balances over time.
Tips
- Confusing the growth rate: Be careful not to misinterpret the data as a decrease (negative growth).
- Miscalculating the growth rate from the provided balances, which could lead to selecting the wrong formula.
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