Problem 89
Question
Investment The time required to double the amount of an investment at an interest rate \(r\) compounded continuously is given by $$ t=\frac{\ln 2}{r} $$ Find the time required to double an investment at \(6 \%, 7 \%\) and \(8 \% .\)
Step-by-Step Solution
Verified Answer
At 6%: 11.55 years, at 7%: 9.90 years, at 8%: 8.66 years.
1Step 1: Identify the Given Information
The problem provides different interest rates: 6%, 7%, and 8%. These need to be converted into decimal form for use in calculations. So, we have \( r = 0.06 \), \( r = 0.07 \), and \( r = 0.08 \).
2Step 2: Apply the Doubling Time Formula
The formula to find the time required to double an investment is \( t = \frac{\ln 2}{r} \). Here, \( \ln 2 \approx 0.693 \). We will apply this formula separately for each interest rate.
3Step 3: Calculate for 6% Interest Rate
Substitute \( r = 0.06 \) into the doubling time formula:\[t = \frac{\ln 2}{0.06} \approx \frac{0.693}{0.06} \approx 11.55 \text{ years}.\]
4Step 4: Calculate for 7% Interest Rate
Substitute \( r = 0.07 \) into the same formula:\[t = \frac{\ln 2}{0.07} \approx \frac{0.693}{0.07} \approx 9.90 \text{ years}.\]
5Step 5: Calculate for 8% Interest Rate
Substitute \( r = 0.08 \) into the formula:\[t = \frac{\ln 2}{0.08} \approx \frac{0.693}{0.08} \approx 8.66 \text{ years}.\]
Key Concepts
Continuous CompoundingExponential GrowthInterest Rate Calculation
Continuous Compounding
Continuous compounding is a concept in finance and math that refers to the idea of applying interest to an investment continuously, rather than periodically.
This means the investment amount grows all the time, instead of at set intervals like monthly or annually. This leads to faster growth over time.
The formula used in continuous compounding is \[ A = P imes e^{rt} \] where:
This means the investment amount grows all the time, instead of at set intervals like monthly or annually. This leads to faster growth over time.
The formula used in continuous compounding is \[ A = P imes e^{rt} \] where:
- \( A \) is the final amount after time \( t \),
- \( P \) is the principal starting amount,
- \( e \) is the base of the natural logarithms, approximately equal to 2.71828,
- \( r \) is the interest rate as a decimal,
- \( t \) is the time elapsed.
Exponential Growth
Exponential growth describes a process where the rate of increase of a value is proportional to the initial size of that value.
In simpler terms, as the amount increases, it grows even faster, forming a sharp upward curve on a graph. This concept is a cornerstone in understanding how investments grow over time with interest.
In the context of finance, an investment grows exponentially when interest is compounded, meaning that both the initial principal and the accumulated interest earn interest in the next compounding period: \[ A(t) = A_0 imes e^{rt} \]where:
In simpler terms, as the amount increases, it grows even faster, forming a sharp upward curve on a graph. This concept is a cornerstone in understanding how investments grow over time with interest.
In the context of finance, an investment grows exponentially when interest is compounded, meaning that both the initial principal and the accumulated interest earn interest in the next compounding period: \[ A(t) = A_0 imes e^{rt} \]where:
- \( A(t) \) is the amount of money accumulated after time \( t \).
- \( A_0 \) is the initial investment amount.
- \( r \) is the rate of growth or return.
Interest Rate Calculation
Interest rate calculation is an essential skill for managing investments effectively. It involves understanding how different rates affect the growth of your funds.
The interest rate defines how much you will earn on your principal amount over a specific period.
When evaluating investments, converting interest rates to decimal form is crucial to calculation accuracy. For example, a 6% interest rate converts to 0.06, 7% to 0.07, and 8% to 0.08.
An important formula for interest rates when it comes to continuous compounding is the doubling time formula: \[ t = \frac{\ln(2)}{r} \]Here, \( \ln(2) \approx 0.693 \), is the natural logarithm of 2. This formula tells you how long it will take for an investment to double at a specific interest rate.
The interest rate defines how much you will earn on your principal amount over a specific period.
When evaluating investments, converting interest rates to decimal form is crucial to calculation accuracy. For example, a 6% interest rate converts to 0.06, 7% to 0.07, and 8% to 0.08.
An important formula for interest rates when it comes to continuous compounding is the doubling time formula: \[ t = \frac{\ln(2)}{r} \]Here, \( \ln(2) \approx 0.693 \), is the natural logarithm of 2. This formula tells you how long it will take for an investment to double at a specific interest rate.
- For 6%, doubling time \( t \approx \frac{0.693}{0.06} \approx 11.55 \) years.
- For 7%, \( t \approx \frac{0.693}{0.07} \approx 9.90 \) years.
- For 8%, \( t \approx \frac{0.693}{0.08} \approx 8.66 \) years.
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