/*! 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 21 How long does it take for money ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 21

How long does it take for money to triple when compounded continuously at \(5 \%\) per year?

Short Answer

Expert verified
It takes approximately \(21.972\) years for the money to triple when compounded continuously at \(5\%\) per year.

Step by step solution

01

Understand the continuous compound interest formula

The formula for continuously compounded interest is: \(A = P e^{rt}\) Where: - A is the final amount after t years - P is the principal (initial) amount - r is the annual interest rate (in decimal form) - t is the time in years - e is the base of the natural logarithm (approximately 2.71828) In our problem, we want to find the time it takes to triple the initial amount. So, the final amount (A) will be three times the principal (P).
02

Set up the equation

We want to find the time it takes for the money to triple, so we can set up the equation as: \(3P = P e^{0.05t}\)
03

Solve for time (t)

First, divide both sides of the equation by P to isolate the exponential term: \(3 = e^{0.05t}\) Now, we need to solve for t. To do this, take the natural logarithm (ln) of both sides of the equation: \(ln(3) = ln(e^{0.05t})\) Using the property of logarithms, we can simplify the right side of the equation: \(ln(3) = 0.05t \cdot ln(e)\) Since ln(e) is equal to 1, we have: \(ln(3) = 0.05t\) Now, divide both sides by 0.05 to get t: \(t = \frac{ln(3)}{0.05}\)
04

Calculate the value of t

Using a calculator, find the value of t: \(t = \frac{ln(3)}{0.05} 鈮� 21.972\) So, it takes approximately 21.972 years for the money to triple when compounded continuously at 5% per year.

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!

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

Find the length of the graph of the function \(f\) defined by $$ f(x)=\sqrt{25-x^{2}} $$ on the interval [0,5] .

About how many years does it take for money to double when compounded continuously at 2\% per year?

Find a number \(r\) such that $$ \left(1+\frac{r}{10^{75}}\right)^{\left(10^{75}\right)} \approx 4 $$

The functions cosh and \(\sinh\) are defined by $$ \cosh x=\frac{e^{x}+e^{-x}}{2} \text { and } \sinh x=\frac{e^{x}-e^{-x}}{2} $$ for every real number \(x .\) For reasons that do not concern us here, these functions are called the hyperbolic cosine and hyperbolic sine; they are useful in engineering. Show that $$ \cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y $$ for all real numbers \(x\) and \(y\).

Estimate the indicated value without using a calculator. $$ e^{0.0013} $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.