Question

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 the advice: Start saving for retirement as early as possible.

Short answer

Short Answer: The balance in the retirement account after 5, 15, 25, and 35 years was calculated using the formula A(t) = 50,000(1.005^(12t) - 1), and the average rate of change between the intervals [5, 15], [15, 25], and [25, 35] was determined. The account balance after investing for 40 years when starting at age 25 was also computed from this formula. To explain the result, the derivative of A(t) with respect to time, dA/dt, was calculated, illustrating the increasing effect of compound interest on the account balance over time, thus emphasizing the importance of starting to save for retirement early.

Step-by-step solution

Calculate account balances after 5, 15, 25, and 35 years

To find the balance in the account after 5, 15, 25, and 35 years, we will substitute the corresponding values of \(t\) into the formula \(A(t) = 50,000(1.005^{12t} - 1)\). So in this exercise we have A(5), A(15), A(25) and A(35)

Calculate the average rate of change in the intervals [5, 15], [15, 25], and [25, 35]

To determine the average rate of change in the account balance over the intervals [5, 15], [15, 25], and [25, 35], we will use the formula: Average rate of change = \(\frac{A(t_2) - A(t_1)}{t_2 - t_1}\) Where \(t_2 > t_1\) and \(A(t_i)\) represents the account balance after \(t_i\) years. We will calculate the average rate of change for each interval mentioned above.

Calculate the balance after 40 years of investing if the plumber started saving at age 25

To find the retirement account balance at age 65 after 40 years of investing, we need to evaluate the function \(A(t)\) with \(t=40\) using the formula: \(A(t) = 50,000(1.005^{12t} - 1)\)

Calculate the derivative of A(t) to explain the results of part (b)

To find the derivative \(\frac{dA}{dt}\) of the account balance with respect to time, we will differentiate the formula \(A(t) = 50,000 \cdot (1.005^{12t} - 1)\) with respect to \(t\). Then, we will use the derivative to explain the surprising result in part (b) and the advice to start saving for retirement early. So we'll apply the chain rule to obtain \(\frac{dA}{dt}\) with respect to \(t\), which will look like this: \(\frac{dA}{dt} = 50,000(1.005^{12t} - 1)'\) Now use the properties of derivatives and the chain rule to calculate the derivative.

Key concepts

Retirement Savings

Retirement savings are an essential aspect of financial planning, ensuring that individuals have sufficient funds to maintain their lifestyle after leaving the workforce. In this scenario, we discuss a self-employed plumber who starts saving $250 per month at the age of 30. He continues this practice until he reaches 65. Given the account offers a 6% interest rate, compounded monthly, this forms a compound interest problem.Compound interest grows at an accelerating rate because interest earned is added to the principal, and future interest calculations include interest on top of prior interest. This feature is highly beneficial for long-term savings like retirement accounts. The formula used here is: \[ A(t) = 50,000(1.005^{12t} - 1) \]. By substituting different values of \( t \) (such as 5, 15, 25, and 35 years), we determine how much money the plumber has accumulated over each period. This approach assists in visualizing growth over time and emphasizes the impact of regular contributions paired with compound interest.

Average Rate of Change

The average rate of change in this context allows us to understand how fast the account balance increases over specific periods. It's essentially the difference in total amount over a time interval divided by that interval's length. For the plumber, we compute the average rate of change over intervals like \[ [5, 15], [15, 25], \text{and} [25, 35] \].Using the formula: \[ \frac{A(t_2) - A(t_1)}{t_2 - t_1} \], where \( A(t_i) \) is the account balance at time \( t_i \). This computation provides insights into how rapidly the savings grow within those periods. Typically in compound interest scenarios, the rate of increase accelerates over time due to the compound nature of the interest, illustrating that returns grow not only on the contributions but also on the accumulated interest.

Derivative of Compound Interest

The derivative of the compound interest formula provides a more nuanced understanding of how the savings grow over time. The derivative, \( \frac{dA}{dt} \), highlights the rate at which the account balance is increasing at any given moment. It is calculated using the chain rule applied to the compound interest formula: \[ A(t) = 50,000(1.005^{12t} - 1) \].By differentiating with respect to \( t \), we analyze changes in the growth rate itself, which helps explain why an early start significantly impacts total savings.The surprising result from part (b) of the exercise, where the plumber saves longer by starting earlier, can be explained through this derivative. As time progresses, the compound effect becomes more pronounced, emphasizing why starting to save early in one’s career dramatically increases final savings. This is due to the longer duration of interest compounding, which allows every dollar saved to benefit from more compounding periods, leading to exponential growth.