Problem 40 Find the distance traveled (to t... [FREE SOLUTION] (2024)

Get started for free

Get started for free Log out

Chapter 2: Problem 40

Find the distance traveled (to three decimal places) from \(t=1\) to \(t=5\)seconds, for a particle whose velocity is given by \(v(t)=t+\ln t .\) (A) 6.000 (B) 1.609 (C) 16.047 (D) 148.413

Short Answer

Expert verified

The answer is (C) 16.047.

Step by step solution

Identify the velocity function and the time interval

The velocity function is given by \(v(t)=t+\ln t .\) And we are given a time interval from \(t=1\) to \(t=5\) seconds.

Setup the Integral

Set up the definite integral of the absolute value of velocity from \(t=1\) to \(t=5\) seconds. Since the velocity function \(v(t)=t+\ln t\) is always increasing in the interval 1 to 5, we do not need to consider the absolute value. The integral then is: \(\int _1^5 v(t) dt\).

Evaluate the Integral

To evaluate this integral \(\int_1^5(t+\ln t) dt\), we can separate the two parts, reduce to the simpler integral of \(t\) and \(\ln(t)\) respectively, then use the fundamental theorem of calculus. The definite integral thus becomes: \(\int _1^5 t dt + \int _1^5 \ln(t) dt \). Each part can be calculated separately. The integral of \(t\) from 1 to 5 is \((1/2)*t^2]_1^5 = 12\) and the integral of \(\ln(t)\) from 1 to 5 is \(t*\ln(t) - t]_1^5 = (5 * \ln(5) -5) - (1 * \ln(1) -1) = 5 * \ln(5) -4\). Add them up, we get \(12 + 5 * \ln(5) -4 = 16.047.\)

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

\(\lim _{x \rightarrow 0} \frac{x \cdot 2^{x}}{2^{x}-1}=\) (A) \(\ln 2\) (B) 1 (C) 2 (D) \(\frac{1}{\ln 2}\)Sea grass grows on a lake. The rate of growth of the grass is \(\frac{d G}{dt}=k G\) where \(k\) is a constant. (a) Find an expression for \(G\) , the amount of grass in the lake (in tons), in terms of \(t\) , the number of years, if the amount of grass is 100 tonsinitially and 120 tons after one year. (b) In how many years will the amount of grass available be 300 tons? (c) If fish are now introduced into the lake and consume a consistent 80 tons/year of sea grass, how long will it take for the lake to be completely free of sea grass?\(\int \sin ^{5}(2 x) \cos (2 x) d x=\) (A) \(\frac{\sin ^{6} 2 x}{12}+C\) (B) \(\frac{\sin ^{6} 2 x}{6}+C\) (C) \(\frac{\cos ^{5} 2 x}{3}+C\) (D) \(\frac{\cos ^{5} 2 x}{6}+C\)Approximate \(\int_{0}^{1} \sin ^{2} x d x\) using the Trapezoid Rule with\(n=4,\) to three decimal places. (A) 0.277 (B) 0.555 (C) 1.109 (D) 2.219\(\lim _{x \rightarrow 0} \frac{8 x^{2}}{\cos x-1}=\) (A) \(\quad-16\) (B) \(\quad-1\) (C) \(\quad 8\) (D) \(\quad 6\)

See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

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.

Sign-up for free

This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept

Privacy & Cookies Policy

Privacy Overview

This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.

Necessary

Always Enabled

Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.

Non-necessary

Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.

SAVE & ACCEPT

Problem 40 Find the distance traveled (to t... [FREE SOLUTION] (2024)

Short Answer

Step by step solution

Identify the velocity function and the time interval

Setup the Integral

Evaluate the Integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

Applied Mathematics

Discrete Mathematics

Logic and Functions

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Privacy Overview

References