Question

Savings plan Beginning at age \(30,\) a self-employed plumber saves \(\$ 250\) per month in a retirement account until he reaches age \(65 .\) The account offers \(6 \%\) interest, compounded monthly. The balance in the account after \(t\) years is given by \(A(t)=50,000\left(1.005^{12 t}-1\right)\) a. Compute the balance in the account after \(5,15,25,\) and 35 years. What is the average rate of change in the value of the account over the intervals \([5,15],[15,25],\) and [25,35]\(?\) b. Suppose the plumber started saving at age 25 instead of age 30\. Find the balance at age 65 (after 40 years of investing). c. Use the derivative \(d A / d t\) to explain the surprising result in part (b) and to explain this advice: Start saving for retirement as early as possible.

Short answer

Question: Compute the account balances after 5, 15, 25, and 35 years for a plumber using the function \(A(t)=50,000\left(1.005^{12 t}-1\right)\). Also, find the average rate of change over the given intervals. Calculate the balance at age 65 if the plumber started saving at age 25 instead of age 30. Finally, explain the result in part (b) using the derivative \(d A / d t\). Answer: After 5, 15, 25, and 35 years, the account balances are calculated by plugging the corresponding years into the given function. The average rate of change for the given intervals can be calculated by dividing the difference in account balances over the difference in time. If the plumber started saving at age 25 instead of age 30, the account balance at age 65 would be found using \(A(40)\). The derivative of the function \(A(t)\) can be used to explain the result in part (b), which represents the rate of change of the balance. The earning power of the account increases with time, implying that starting to save earlier results in a larger account balance in the end.

Step-by-step solution

Compute balances after 5, 15, 25, and 35 years

Plug the given years into the function \(A(t)=50,000\left(1.005^{12 t}-1\right)\) to get their respective balances. After 5 years: \(A(5)=50,000\left(1.005^{12 \cdot 5}-1\right)\) After 15 years: \(A(15)=50,000\left(1.005^{12 \cdot 15}-1\right)\) After 25 years: \(A(25)=50,000\left(1.005^{12 \cdot 25}-1\right)\) After 35 years: \(A(35)=50,000\left(1.005^{12 \cdot 35}-1\right)\)

Calculate the average rate of change

Find the average rate of change for the given intervals. For the interval [5, 15]: \(\frac{A(15) - A(5)}{15-5}\) For the interval [15, 25]: \(\frac{A(25) - A(15)}{25-15}\) For the interval [25, 35]: \(\frac{A(35) - A(25)}{35-25}\) #b. Balance after starting at age 25#

Find the balance at age 65 having started saving at age 25

We need to find the balance after 40 years of investing. \(A(40)=50,000\left(1.005^{12 \cdot 40}-1\right)\) #c. Explain the results using the derivative#

Calculate the derivative \(dA/dt\)

Find the derivative of the function \(A(t)=50,000\left(1.005^{12 t}-1\right)\). \(d A / d t = d / d t \left[50,000\left(1.005^{12 t}-1\right)\right]\)

Explain the surprising result and advice

Using the derivative \(d A / d t\), discuss the result in part (b) and the advice to start saving for retirement as early as possible. Consider that the derivative represents the rate of change of the balance and thus reflects the earning power of the account. Since the function is always increasing, starting earlier results in a larger balance in the retirement account.

Key concepts

Compound Interest

Compound interest is a fundamental financial concept that involves earning interest on both the initial principal and the accumulated interest from previous periods. This concept is crucial when planning for retirement savings, as seen in the exercise where a plumber invests a fixed amount monthly at an interest rate of 6% compounded monthly.

• The formula for compound interest is often given as: \[ A(t) = P \left(1 + \frac{r}{n}\right)^{nt} \]where:
  • \(P\) is the principal amount (initial investment).
  • \(r\) is the annual nominal interest rate (as a decimal).
  • \(n\) is the number of times the interest is compounded per year.
  • \(t\) is the time the money is invested for (in years).
  • \(A(t)\) is the amount of money accumulated after n years, including interest.

In our case, the function given \( A(t) = 50,000(1.005^{12t} - 1) \), represents the compounded growth over multiple years from saving $250 each month. The value of 1.005 accounts for the monthly compounding of the 6% annual interest.

Rate of Change

The rate of change is a measure used to understand how a quantity changes over time. In the context of retirement savings, the rate of change helps us understand how much the account balance is growing during different periods.

• To find the average rate of change, we use:\[\frac{A(b) - A(a)}{b - a}\]where \(A(b)\) and \(A(a)\) are the account balances at times \(b\) and \(a\), respectively.

In our exercise, we calculated the average rate of change over intervals like [5, 15], [15, 25], and [25, 35] years. This gives us an important insight into how rapidly the account value is increasing during these specific times.

Financial Planning

Effective financial planning requires a combination of understanding the nuances of interest, regular savings, and time. Planning for retirement savings at a young age can drastically improve financial security later in life.

• The longer the investment period, the more significant the impact of compound interest.
• Committing to a savings plan with consistent contributions over many years leads to exponential growth due to compound interest.
• Starting savings early allows the money to work harder and multiply over an extended period.

By using the plumber's example, financial planners can illustrate the benefits of early savings significantly affecting the final balance by age 65, thus securing a stronger financial foundation for retirement.

Derivative Analysis

Derivative analysis involves understanding how a function's output changes concerning its inputs. The derivative of a function provides the rate at which the function's value is changing at any given point.

• The derivative \( \frac{dA}{dt} \) of the savings function \( A(t) = 50,000(1.005^{12t} - 1) \) tells us how the retirement account's balance is changing with time.
• A positive and increasing derivative indicates a growing account balance.

In the exercise, derivative analysis helps explain why beginning to save at age 25 rather than 30 resulted in a larger balance: the earlier you start, the longer the money benefits from compounding. Thus, early investment leverages these compounding effects, allowing the savings to grow at an accelerating pace. This analysis provides compelling evidence to reinforce the advice of starting to save for retirement as early as possible.