Problem 66 - Precalculus : Building Concepts and Connections

Question

Determine how long it takes for the given investment to double if \(r\) is the interest rate and the interest is compounded continuously. Assume that no withdrawals or further deposits are made. Initial amount: \(\$ 3800 ; r=5.8 \%\)

Step-by-Step Solution

Answer

It will take approximately 12 years for the investment to double.

Step 1: Understand the problem

We need to find the time \(t\) it takes for an initial investment of $3800 to double, given an interest rate of 5.8% compounded continuously. This can be achieved by setting \(A=2*P\), and \(r=5.8/100\), and replacing these in the formula for continuous compounding.

Step 2: Isolate the unknown

So, we write the formula for continuous compounding interest with the known variables: \(7600 = 3800 * e^{0.058t}\). We can isolate \(t\) by first dividing each side by 3800, which gives us \(2=e^{0.058t}\). Taking the natural logarithm (ln) on both sides, we have \(ln(2)=0.058t\). When we isolate \(t\), we find that \(t=ln(2)/0.058.\)

Step 3: Calculate the unknown

By inserting the known values into the equation, we can find the time value \(t\). This gives \(t=ln(2)/0.058 = 12\,years\).

Key Concepts

Exponential Growth

Exponential growth describes the phenomena in many natural and financial systems where quantities increase at a rate proportional to their current value. This means the bigger the quantity gets, the faster it grows. In the context of an investment, exponential growth occurs when the interest earned on the principal amount is reinvested, earning more interest in the next period. This concept is crucial when discussing compound interest, especially when it is compounded continuously.

For our exercise example where an investment doubles over time, the final amount after time period t can be described by the formula: \[ A = Pe^{rt} \]where,

• A is the amount of money accumulated after time t, including interest.
• P is the principal amount (the initial amount of money).
• r is the annual interest rate (in decimal form).
• t is the time the money is invested for.
• e is the base of the natural logarithm, reflecting continuous compounding.

In the example, the initial investment is doubled, so if P is \(3800, A is set to \)7600. The growth to this amount is not linear; it accelerates over time due to the continuous compounding effect. To solve for the time it takes to double, we work out the algebra involving these exponential equations.

Natural Logarithm

The natural logarithm, often represented by ln, is the logarithm to the base e, where e is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm is the inverse function to the exponential function, meaning it undoes the effect of an exponentiation of e.

For example, if we have an equation \[e^x = b\],then by taking the natural logarithm of both sides we get\[ln(e^x) = ln(b)\],which simplifies to\[x = ln(b)\].This property is particularly useful for solving problems involving continuous compounding interest, as it allows us to isolate the exponent (which usually contains the period t) and solve for it, as demonstrated in the step-by-step solution above. When we had \[2 = e^{0.058t}\],taking the natural logarithm of both sides made it possible to solve for t by isolating it on one side of the equation.

Time Value of Money

The time value of money is a financial concept which states that a sum of money is worth more now than the same sum in the future due to its potential earning capacity. This principle is foundational to finance and investment strategies and is why interest is charged or earned on money. The idea is that money can earn interest, so it is better to receive money now rather than later.

Continuously compounded interest is a powerful example of the time value of money. It shows how money can grow over time at an exponential rate. Our interest calculation takes into account that the interest is not just calculated periodically, but at every possible instant, which results in the fastest growth rate possible for your money. The formula applied in our exercise, where \[ A = Pe^{rt} \],encapsulates this concept by indicating how time (t) interacts with the principal amount (P) and the rate (r) to grow the investment exponentially.