/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 An amount of invested money is s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 13

An amount of invested money is said to draw interest compounded continuously if the amount of money increases at a rate proportional to the amount present. Suppose \(\$ 1000\) is invested and draws interest compounded continuously, where the annual interest rate is \(6 \%\) (a) How much money will be present 10 years after the original amount was invested? (b) How long will it take the original amount of money to double?

Short Answer

Expert verified
In part (a), the amount of money after 10 years is approximately $1822.10. In part (b), it takes about 11.55 years for the investment to double.

Step by step solution

01

Part (a): Calculate the invested amount after 10 years

The formula for continuous compound interest is \(A = P * e^{rt}\). Given the initial investment amount is \(1000$ (P = 1000)\), the annual interest rate is \(6 \%\) (r = 0.06), and the time period is 10 years (t = 10). We will substitute these values into the formula and find the final amount (A). \(A = 1000 * e^{(0.06)(10)}\) Now, we will calculate A: \(A \approx 1000 * e^{0.6} \approx 1000 * 1.8221 \) \(A \approx 1822.10\) There will be approximately $1822.10 after 10 years.
02

Part (b): Calculate the time required to double the invested amount

To find the time (t) it takes for the initial investment to double, we will set A equal to twice the initial investment (2P) and solve for t in the continuous compound interest formula. \(2P = P * e^{rt}\) First, we will divide both sides by P: \(2 = e^{rt}\) Now, we need to solve for t. To do this, take the natural logarithm of both sides: \(ln(2) = ln(e^{rt})\) Use the property of logarithms to bring down the exponent: \(ln(2) = rt * ln(e)\) Since \(ln(e) = 1\), this simplifies to: \(ln(2) = rt\) Now, we will substitute the given annual interest rate (r = 0.06) and solve for t: \(ln(2) = 0.06t\) \(t = \frac{ln(2)}{0.06}\) Finally, we will calculate the value of t: \(t \approx \frac{0.6931}{0.06} \approx 11.55\) It will take approximately 11.55 years for the invested amount to double.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Interest Rates
Interest rates are a fundamental concept in finance and investing. When someone refers to an interest rate, they are usually talking about the percentage charge on borrowed money or earned on invested funds over a period of time. There are different types of interest rates, but in this article, we'll focus on those relevant to compound interest.
  • Fixed vs. Variable Rates: Fixed rates stay the same throughout the life of the loan or investment, while variable rates may fluctuate based on market conditions.
  • Nominal vs. Effective Rates: The nominal rate is the stated rate, while the effective rate accounts for the effect of compounding. The effective rate is typically higher if interest is compounded frequently.
  • Annual Percentage Rate (APR): This rate is often used in the context of loans and includes fees or additional costs associated with the transaction.
In the context of continuous compound interest, the interest rate is applied continuously, meaning that at every possible moment, the interest is being calculated and added to the principal. This results in an exponential increase over time, making understanding these rates crucial for financial planning and investment strategies.
Exponential Growth
Exponential growth occurs when the rate of growth of a mathematical function is proportional to the current value, leading to growth with increasing speed. This is a common phenomenon in finance, particularly in compound interest scenarios. Consider the formula for continuous compound interest: \[A = P * e^{rt}\]Where:
  • A is the amount of money accumulated after time t.
  • P is the principal amount (initial investment).
  • e is the base of the natural logarithms, approximately equal to 2.71828.
  • r is the annual interest rate (in decimal form).
  • t is the time in years.
Exponential growth underscores how investments grow at an increasing rate due to the effect of compounding. The longer money is invested, the faster it grows, which is a key reason why investing early can significantly increase future wealth. This growth pattern is not linear, which is why the future value of investments with compound interest can be surprisingly large.
Investment Doubling Time
Investment doubling time is a simple concept that measures how long it takes for an investment's value to double at a given constant interest rate. Understanding this concept can greatly aid in setting realistic financial goals and understanding the power of compound interest over time. In continuous compounding, doubling time can be calculated using the formula:\[t = \frac{\ln(2)}{r}\]Where:
  • t is the time required to double the investment.
  • \ln(2) is the natural logarithm of 2, approximately equal to 0.6931.
  • r is the annual interest rate in decimal form.
By substituting 6% (or 0.06 in decimal form) into the formula, we can see that it takes approximately 11.55 years for the investment to double. This method of calculating doubling time is particularly helpful for investors looking to evaluate how different interest rates affect the growth of their investments. Observing how doubling time decreases with increasing interest rates can encourage a more aggressive investment strategy.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A certain chemical is converted into another chemical by a chemical reaction. The rate at which the first chemical is converted is proportional to the amount of this chemical present at any instant. Ten percent of the original amount of the first chemical has been converted in 5 min. (a) What percent of the first chemical will have been converted in 20 min? (b) In how many minutes will \(60 \%\) of the first chemical have been converted?

Assume that the population of a certain city increases at a rate proportional to the number of inhabitants at any time. If the population doubles in 40 years, in how many years will it triple?

A case of canned milk weighing \(24 \mathrm{lb}\) is released from rest at the top of a plane metal slide which is \(30 \mathrm{ft}\) long and inclined \(45^{\circ}\) to the horizontal. Air resistance (in pounds) is numerically equal to one-third the velocity (in feet per second) and the coefficent of friction is \(0.4\). (a) What is the velocity of the moving case 1 sec after it is released? (b) What is the velocity when the case reaches the bottom of the slide?

Find the orthogonal trajectories of each given family of curves. In each case sketch several members of the family and several of the orthogonal trajectories on the same set of axes. $$ x-y=c x^{2} $$

Assume that the rate of change of the human population of the earth is proportional to the number of people on earth at any time, and suppose that this population is increasing at the rate of \(2 \%\) per year. The 1979 World Almanac gives the 1978 world population estimate as 4,219 million; assume this figure is in fact correct. (a) Using this data, express the human population of the earth as a function of time. (b) According to the formula of part (a), what was the population of the earth in \(1950 ?\) The 1979 World Almanac gives the 1950 world population estimate as 2,510 million. Assuming this estimate is very nearly correct, comment on the accuracy of the formula of part (a) in checking such past populations. (c) According to the formula of part (a), what will be the population of the earth in \(2000 ?\) Does this seem reasonable? (d) According to the formula of part (a), what was the population of the earth in \(1900 ?\) The 1979 World Almanac gives the 1900 world population estimate as 1,600 million. Assuming this estimate is very nearly correct, comment on the accuracy of the formula of part (a) in checking such past populations. (c) According to the formula of part (a), what will be the population of the earth in \(2100 ?\) Does this seem reasonable?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.