Problem 49 - Calculus: Graphical, Numerical, Algebraic

Question

Multiple Choice A bank account earning continuously compounded interest doubles in value in 7.0 years. At the same interest rate, how long would it take the value of the account to triple? (A) 4.4 years (B) 9.8 years (C) 10.5 years (D) 11.1 years (E) 21.0 years

Step-by-Step Solution

Answer

The time it would take for the value of the account to triple at the same interest rate is approximately 11.1 years.

Step 1 - Find the expression for r

From the information given, we know that the money doubles in size in 7 years. This means that \(A = 2P\). Inputting these into the formula for continuously compounded interest, \(2P = Pe^{7r}\). Simplifying this equation leads to \(2 = e^{7r}\). Taking the natural logarithm of both sides, \(ln(2) = 7r\), which results in an expression for r being \(r = \frac{ln(2)}{7}\).

Step 2 - Substitute r into the equation and find t

Now, we can substitute the value of r into the equation \(A = Pe^{rt}\) to find the time it takes for the money to triple in size \(i.e.\, A = 3P\). The equation would then be \(3P = Pe^{t \frac{ln(2)}{7}}\). Simplifying this, we get \(3 = e^{t \frac{ln(2)}{7}}\). Now, taking the natural logarithm on both sides, we have \(ln(3) = t \frac{ln(2)}{7}\). This simplifies to \(t = \frac{7ln(3)}{ln(2)}\)

Step 3 - Calculate the value of t

To find t, substitute ln(2) and ln(3) with their values: t = \(frac{7*1.0986}{0.6931}\) which will result in t ≈ 11.1

Key Concepts

Exponential Growth

Exponential growth describes a process where quantities increase at a rate proportional to their current value, which can be observed in various natural and financial contexts. Such growth is characterized by the presence of a constant base raised to a variable exponent in mathematical terms, typically represented as \( A = Pe^{rt} \), where \( P \) is the initial amount, \( r \) is the growth rate, and \( t \) is the time period involved. In the realm of finance, this formula is used for continuously compounded interest, indicating that the investment grows exponentially over time as interest is calculated on an instantly compounded basis. The fascinating aspect of exponential growth in investments is that while the time to double an investment is fixed for a specific interest rate, the time to triple is not simply a multiple of the 'doubling time' but rather calculated using the same principles in a slightly different context, emphasizing the non-linearity of exponential functions.

Natural Logarithm

The natural logarithm or \(\ln(x)\) is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. This function is the inverse of the exponential function, which means that \(\ln(e^x) = x\) and conversely, \(e^{\ln(x)} = x\). In the context of continuously compounded interest, when we have an equation like \(2 = e^{7r}\), taking the natural logarithm of both sides leads to \(\ln(2) = 7r\). This ability to linearize the relationship between variables in exponential equations is a powerful tool. It allows us to isolate the growth rate (\