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Chapter 4: Problem 111

Solve each problem. When needed, use 365 days per year and 30 days per month. Working by the Hour One million dollars is deposited in an account paying \(6 \%\) compounded continuously. a. What amount of interest will it earn in its first hour on deposit? Hint One hour is \(1 / 8760\) of a year. b. What amount of interest will it earn during its 500 th hour on deposit? Hint Subtract the amount for \(t=499\) from the amount for \(t=500\)

Short Answer

Expert verified
a) Interest for the first hour: 6.849 b) Interest for the 500th hour: 6.833

Step by step solution

01

- Understand the continuous compounding formula

The formula for continuous compounding is given by: \[A = P e^{rt} \] Where: - \(A\) is the amount of money accumulated after a time period \(t\) - \(P\) is the principal amount - \(r\) is the annual interest rate - \(t\) is the time the money is invested or borrowed for, in years - \(e\) is the base of the natural logarithm
02

- Calculate the interest for the first hour (a)

Given: - \( P = 1,000,000 \) - \( r = 0.06 \) - \( t = \frac{1}{8760} \)Use the continuous compounding formula: \[ A = 1,000,000 \times e^{0.06 \times \frac{1}{8760}} \] Calculate the exponent: \[ 0.06 \times \frac{1}{8760} = 6.849315068 \times 10^{-6} \] Then \[ A = 1,000,000 \times e^{6.849315068 \times 10^{-6}} \] Using a calculator to evaluate, \( e^{6.849315068 \times 10^{-6}} \) is approximately 1.000006849. Therefore: \[ A = 1,000,000 \times 1.000006849 \ A = 1,000,006.849 \] The amount of interest earned in the first hour is: \[ 1,000,006.849 - 1,000,000 = 6.849 \]
03

- Calculate the interest for the 500th hour (b)

For the amount at \(t = 500\) hours: \[ t = \frac{500}{8760} \] Use the formula: \[ A_{500} = 1,000,000 \times e^{0.06 \times \frac{500}{8760}} \] Calculate the exponent: \[ 0.06 \times \frac{500}{8760} = 0.003424657534 \] Therefore \[ A_{500} = 1,000,000 \times e^{0.003424657534} \] Approximating: \( e^{0.003424657534} \) is approximately 1.003431144. Hence: \[ A_{500} = 1,003,431.144 \] For the amount at \(t = 499\) hours: \[ t = \frac{499}{8760} \] Use the formula: \[ A_{499} = 1,000,000 \times e^{0.06 \times \frac{499}{8760}} \] Calculate the exponent: \[ 0.06 \times \frac{499}{8760} = 0.003417808219 \] Then \[ A_{499} = 1,000,000 \times e^{0.003417808219} \] Approximating: \( e^{0.003417808219} \) is approximately 1.003424311 . Hence: \[ A_{499} = 1,003,424.311 \] The interest earned during the 500th hour is the difference: \[ A_{500} - A_{499} = 1,003,431.144 - 1,003,424.311 = 6.833 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Compounded Interest
Compounded interest is the process where the interest earned over time is reinvested into the principal amount.
This results in earning more interest in subsequent periods.
Continuous compounding is a special case where interest is added an infinite number of times in each period.
This leads to faster growth compared to other compounding methods like monthly or yearly compounding.
Using continuous compounding, the amount of money in an account grows according to the formula:
\[A = P e^{rt}\]
Here,
  • A is the final amount after time period t
  • P is the initial principal amount
  • r is the annual interest rate (as a decimal)
  • t is the time the money is invested (in years)
  • e is the base of the natural logarithm, approximately equal to 2.71828
Remember, compounding more frequently results in more interest earned, and continuous compounding is the most frequent compounding possible.
Exponential Functions
Exponential functions involve the exponentiation of a constant base.
In continuous compounding, we use the exponential function with a base of e.
The general form of the function is \[f(x) = a \times e^{bx} \]
Here, a and b are constants, and x is the exponent.
  • Exponential growth occurs when the base is greater than 1, like in our formula where the base is e. This illustrates how money grows over time.
  • Exponential decay occurs when the base is between 0 and 1.
In our example, the amount of money grows exponentially over time due to continuous compounding. Calculating the exponent in the formula is crucial:
For example, when calculating the amount of interest earned in the first hour, the exponent \[0.06 \times \frac{1}{8760} \] is very small, leading to a slight increase in the additive exponent, resulting in a slightly larger final amount.
Natural Logarithm
A natural logarithm is the logarithm to the base e, denoted as \[\text{ln}(x)\].
It is the inverse function of the exponential function with base e.
This means if \[e^y = x\], then \[\text{ln}(x) = y\].Natural logarithms are useful in solving equations involving continuous compounding.
As any exponential equation with base e can be resolved using natural logarithms:
  • If we need to solve for time t in the continuous compounding formula \[A = P e^{rt}\], we take the natural logarithm of both sides.
This provides insight into reversing the growth process:
From our continuous compounding formula, taking natural logarithms simplifies exponent manipulation,reiterating the relationship between exponential growth and logarithms.
Understanding these concepts provides more clarity on how interest accumulates over time with continuous compounding.

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Most popular questions from this chapter

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