Problem 30
Question
The amount of money, \(A(t),\) in a savings account that pays \(6 \%\) interest, compounded quarterly for \(t\) years, when an initial investment of \(\$ 2000\) is made, is given by $$ A(t)=2000(1.015)^{4 t} $$ a) Find \(A(3)\). b) Find \(A(5)\) c) Find \(A(5)-A(3)\). d) Find \(\frac{A(5)-A(3)}{5-3}\). What rate of change does this represent?
Step-by-Step Solution
Verified Answer
A(3) and A(5) are the amounts after 3 and 5 years, respectively. The average rate of change represents the average increase in savings per year over 3 to 5 years.
1Step 1: Understand the Function
The function given is \( A(t) = 2000(1.015)^{4t} \). This represents the amount of money in a savings account over \( t \) years, compounded quarterly, with an annual interest rate of \( 6\% \). The term \( 1.015 \) is obtained from the quarterly interest rate \( (\frac{0.06}{4} = 0.015) \).
2Step 2: Calculate A(3)
To find \( A(3) \), substitute \( t = 3 \) into the function: \[ A(3) = 2000(1.015)^{4\times3} = 2000(1.015)^{12} \] Using a calculator, compute \( (1.015)^{12} \) and then multiply by 2000 to find \( A(3). \)
3Step 3: Calculate A(5)
Substitute \( t = 5 \) in the function to find \( A(5) \): \[ A(5) = 2000(1.015)^{4\times5} = 2000(1.015)^{20} \] Calculate \( (1.015)^{20} \) and multiply the result by 2000 to get \( A(5). \)
4Step 4: Calculate A(5) - A(3)
Subtract \( A(3) \) from \( A(5) \) to find the difference: \[ A(5) - A(3) = 2000 \times (1.015)^{20} - 2000 \times (1.015)^{12} \] Calculate the expressions for each \( A(t) \) and then find the difference.
5Step 5: Calculate the Average Rate of Change
To find the average rate of change between \( t = 3 \) and \( t = 5 \), use the formula: \[ \frac{A(5)-A(3)}{5-3} = \frac{2000 \times (1.015)^{20} - 2000 \times (1.015)^{12}}{2} \] Calculate the difference and divide by 2. The result represents the average rate of change of the amount of money over the given period.
Key Concepts
Savings AccountInterest Rate CalculationAverage Rate of Change
Savings Account
A savings account is a secure and reliable way to keep your money while earning some extra through interest. When you deposit money in a savings account, the bank pays you interest for using your funds. This process allows your money to grow over time without additional effort from your side. Many savings accounts offer compound interest, which means you earn interest not just on your initial deposit but also on the interest that accumulates over time. This helps your savings grow faster.
Compound interest is particularly beneficial for long-term savings plans, as it can significantly increase your savings over the years. Therefore, choosing a savings account with a good interest rate and understanding how the interest compounds can make a big difference in how much money you can save.
Compound interest is particularly beneficial for long-term savings plans, as it can significantly increase your savings over the years. Therefore, choosing a savings account with a good interest rate and understanding how the interest compounds can make a big difference in how much money you can save.
Interest Rate Calculation
To understand how much interest you will earn on a savings account, you need to know how the interest is calculated. Interest can be calculated in different ways, one of which is through compound interest. In our exercise, the interest is compounded quarterly. This means that the interest is calculated and added to your account balance four times a year. Each time this interest is added, it also begins to earn interest itself.
In the exercise, the annual interest rate is 6%, but since the interest compounds quarterly, the quarterly interest rate is 1.5%, which is calculated by dividing the annual rate by four: \[ \text{Quarterly Interest Rate} = \frac{6\%}{4} = 1.5\% \]This quarterly rate is then used in the formula to calculate how the savings grow over time: \[ A(t) = 2000(1.015)^{4t} \]This formula derives the interest multiplier, 1.015, from the quarterly interest rate, showing how the savings increase each quarter.
In the exercise, the annual interest rate is 6%, but since the interest compounds quarterly, the quarterly interest rate is 1.5%, which is calculated by dividing the annual rate by four: \[ \text{Quarterly Interest Rate} = \frac{6\%}{4} = 1.5\% \]This quarterly rate is then used in the formula to calculate how the savings grow over time: \[ A(t) = 2000(1.015)^{4t} \]This formula derives the interest multiplier, 1.015, from the quarterly interest rate, showing how the savings increase each quarter.
Average Rate of Change
The average rate of change is a mathematical concept that lets us see how one quantity changes, on average, over a specific interval of time. In terms of money in a savings account, it tells us how fast our savings are growing between two points in time.
In the given exercise, we calculate the average rate of change between the third and fifth year:\[ \frac{A(5)-A(3)}{5-3} \]Substituting the values, it becomes:\[ \frac{2000 \times (1.015)^{20} - 2000 \times (1.015)^{12}}{2} \]This calculation shows us how much the amount in the account grows, on average, each year between the third and fifth year. Understanding the average rate of change can help plan financial goals and see if your savings strategy is on target. Whether saving for a future expense, retirement, or education, knowing how your money grows over time is crucial for effective financial planning.
In the given exercise, we calculate the average rate of change between the third and fifth year:\[ \frac{A(5)-A(3)}{5-3} \]Substituting the values, it becomes:\[ \frac{2000 \times (1.015)^{20} - 2000 \times (1.015)^{12}}{2} \]This calculation shows us how much the amount in the account grows, on average, each year between the third and fifth year. Understanding the average rate of change can help plan financial goals and see if your savings strategy is on target. Whether saving for a future expense, retirement, or education, knowing how your money grows over time is crucial for effective financial planning.
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