Problem 102 - College Algebra and Calculus: An Applied Approach

Question

Find the time required for a \(\$ 1000\) investment to double at interest rate \(r\), compounded continuously.\(r=0.0875\)

Step-by-Step Solution

Answer

To find the time it would take for a $1000 investment to double at an 0.0875 interest rate compounded continuously, calculate \( t = ln(2) / 0.0875 \). It yields approximately \(t \approx 7.93\) years.

Step 1: Identify the Knowns

From the problem, we know that the initial principal amount (P) is $1000, the interest rate (r) is 0.0875, and the final amount (A) is double the initial investment, meaning A = $2000.

Step 2: Applying the Formula

The continuous compound interest formula is \(A = P e^{rt}\). In this case, we are asked to solve for 't', so we first substitute the known values of P, A and r into the equation: \(2000 = 1000 * e^{0.0875t}\).

Step 3: Solving for t

Divide both sides of equation by 1000, we get \(2 = e^{0.0875t}\). To solve for t, we take the natural log (ln) of both sides: ln(2) = \(0.0875t * ln(e)\), since \(ln(e) = 1\), the equation simplifies to \(0.0875t = ln(2)\). Thus, \( t = ln(2) / 0.0875 \).

Step 4: Calculating the Final Value

By calculating this last equation, we can get the result for t.

Key Concepts

Exponential Growth

In the world of finance and investment, exponential growth is a term used to describe an increase at a consistent rate over time. While linear growth adds the same amount each period, exponential growth involves a steady percentage increase, making it multiply rapidly as time goes on. It is often expressed using the formula:
\[ A = P e^{rt} \]

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (the initial amount of money).
• \(r\) is the rate of interest per period.
• \(t\) is the time the money is invested for in years.
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828.

The notion of exponential growth is crucial in understanding how quickly investments can increase in value over time. For instance, when compounded continuously, this formula precisely calculates how money grows at every instant, reflecting real-world scenarios where interest could be accruing every moment.

Natural Logarithm

The natural logarithm, often denoted as \(ln\), is a special type of logarithm with the base \(e\), where \(e\) is approximately 2.71828. It is a commonly used method to solve equations involving exponential growth, especially those containing the \(e\) constant, like in continuous compounding. In our exercise, we applied the following steps:

• Starting with \(2 = e^{0.0875t}\), to find \(t\), we took the natural logarithm on both sides to get \(ln(2) = 0.0875t \,ln(e)\).
• Since \(ln(e) = 1\), it simplifies the equation to \(0.0875t = ln(2)\).
• We then isolated \(t\) by dividing both sides by 0.0875. This gives us \(t = ln(2) / 0.0875\).

The natural logarithm is powerful in simplifying exponents, turning multiplicative processes into additive ones, thus making it easier to solve such problems.
In general, the natural logarithm function helps us unveil how certain processes grow or decay exponentially. It is ideal for calculating timeframes in scenarios involving continuous compounding interest, population growth, or decay in physics and biology.

Investment Doubling Time

Investment doubling time refers to the time required for an investment to grow to twice its original size. A classic way to estimate this is by using the Rule of 72.
However, for precise calculation, especially for continuous compounding as in the exercise, we rely on the formula:
\[ t = \frac{ln(2)}{r} \]
In our textbook example:

• The investment is \(1000 and we want to know how long it takes to reach \)2000.
• Interest rate \(r\) is 0.0875 and employing the formula above, we substitute \(ln(2)\) divided by \(0.0875\) to get \(t\).

This formula provides us with an exact duration by inputting the logarithmic factor, reflecting continuous compounding rather than periodic interest rates. Understanding this helps investors plan better by providing accurate timelines for growth targets, critical for financial strategies and future planning.