/*! 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 3 How much would you need to depos... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 3

How much would you need to deposit in a bank account paying \(4 \%\) annual interest compounded continuously so that at the end of 10 years you would have \(\$ 10,000 ?\)

Short Answer

Expert verified
The initial deposit required to have a balance of $10,000 at the end of 10 years with a 4% annual interest rate compounded continuously is approximately $6,704.04.

Step by step solution

01

Identify the formula for continuous compounding

The formula for continuous compounding is given by: \(A=P\cdot e^{rt}\) where - A is the final amount after time t - P is the principal (initial deposit) - r is the annual interest rate (as a decimal) - t is the time (in years) - e is the base of the natural logarithm (approximately equal to 2.71828)
02

Plug in the known values

We know the final amount A ($10,000), the annual interest rate r (0.04), and the time t (10 years). We want to find the initial deposit P. Plugging these values into the formula, we get: \(10,000=P\cdot e^{(0.04)(10)}\)
03

Solve for P

To find the initial deposit P, we need to isolate P on one side of the equation: \(P=\frac{10,000}{e^{(0.04)(10)}}\) Now, compute the value of the denominator using a calculator: \(P=\frac{10,000}{e^{0.4}}\approx \frac{10,000}{2.71828^{0.4}}\) Using a calculator, we find that: \(P\approx \frac{10,000}{1.49182} \)
04

Calculate the initial deposit

Now, divide $10,000 by the value we obtained in the previous step: \(P\approx 6,704.04\)
05

Interpret the result

The initial deposit required to have a balance of \(10,000 at the end of 10 years with a 4% annual interest rate compounded continuously is approximately \)6,704.04.

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

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 $$ (\cosh x+\sinh x)^{t}=\cosh (t x)+\sinh (t x) $$ for all real numbers \(x\) and \(t\).

Suppose one bank account pays \(3 \%\) annual interest compounded once per year, and a second bank account pays \(4 \%\) annual interest compounded continuously. If both bank accounts start with the same initial amount, how long will it take for the second bank account to contain \(50 \%\) more than the first bank account?

Estimate the value of $$ \left(1+\frac{3}{10^{100}}\right)^{\left(10^{100}\right)} $$. [Your calculator will be unable to evaluate directly the expressions in this exercise and the next five exercises. Thus you will need to do more than button pushing for these exercises.]

(a) Show that $$ 1.01^{100}

Suppose the amount of the world's computer hard disk storage increases by a total of \(200 \%\) over a four-year period. What is the continuous growth rate for the amount of the world's hard disk storage?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

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.