💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Stuck on your homework problem? This step-by-step video should help.

Try Numerade Free for 30 Days

Like

Report

Freshwater is flowing into a brine solution, with an equal volume of mixed solution flowing out. The amount of salt in the solution decreases, but more slowly as time increases. Under certain conditions, the time rate of change of mass of salt (in $\mathrm{g} / \mathrm{min}$ ) is given by $-1 / \sqrt{t+1}$. Find the mass $m$ of salt as a function of time if 1000 g were originally present. Under these conditions, how long would it take for all the salt to be removed?

$m=1002-2 \sqrt{t+1}, 2.51 \times 10^{5} \mathrm{min}$

Calculus 2 / BC

Chapter 26

Applications of Integration

Section 1

Applications of the Indefinite Integral

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

04:21

A brine solution of salt f…

02:40

19:40

07:52

Salt Concentration A tank …

03:48

A tank contains 1000 $\mat…

05:52

Decomposition of Salt in W…

04:28

A tank contains 1000 L of …

05:16

03:50

A $1000$ L tank contains 5…

03:26

So we're told D. M. D. T. Which is the rate of change of the massive salt in grams per minute. Um is equal to negative one divided by T plus one to the one off power. We're also told when T is equal to zero R. Mass is equal to 1000 g. So what we're gonna do to figure out this mass equation is just find this integral down here which is the integral of negative one divided by T plus 1 to 1 off power. Or the integral of our rate of change equation. And that will be equal to our mass equation. So this would be equal to negative one, multiplied by the integral of T. Plus one To the negative 1/2 power. And all I did here was take out the negative one and move this ti plus one in the denominator to the numerator. And so now we can do is we can integrate using substitution. So we can say U. Is equal to T. Plus one then do you? It's equal to one. So are integral. Would now be equal to negative one, multiplied by the integral of you. So the negative 1/2 power do you? So we can integrate this by adding one to the exponent and then dividing by the resulting exponents. So this would be equal to -1, multiplied by You to the 1/2 power. And then dividing by one half is the same as multiplying by two. So we can just multiply by two. And then we also have plus C. And this negative one is just going to distribute to this. You um term it's not going to distribute to the C. Term Since -1 was originally inside of our integral. So now what we can do is simplify this a little bit. We can plug um T plus one in for you. So we have this is equal to negative two, multiplied by T plus one To the 1/2 power. And this is plus C. And so this is equal to our mass equation. Now we can use the fact that our master is equal to 1000 when T. Is equal to zero to figure out what C. Is. So we set 1000 equal to negative two, multiplied by zero plus one to the one half power, pussy zero plus one to the one half power is just one. So now we have 1000 is equal to -2, pussy mm. So then C. is equal to 1002. So now we have our full mass equation. M is equal to -2, Multiplied by T Plus one. It's the one half power Plus 1002. And so we want to figure out how long it's gonna take for all of the salt to no longer be in our solution. So that would mean m is equal to zero. So we just set em equal to zero and solve for T. So we have zero is equal to negative two, multiplied by T plus one to the one of power Plus 1002. You can -1002 under both sides. So yeah -1002. It's equal to negative two times T plus one to the one of power. We can divide both sides by -2. So we get 501, sequel to T plus one To the 1/2 power. And how we can square both sides to get rid of this exponent. So 501 Squared is equal to 251,000 and one. So this is equal to T plus one And how we can -1 over. So we get T is equal to 200 and 51,000 And this is in minutes, so it's going to take 251,000 minutes for the salt to no longer be in our solution.

View More Answers From This Book

Find Another Textbook

In mathematics, integration is one of the two main operations in calculus, w…

In grammar, determiners are a class of words that are used in front of nouns…

A brine solution of salt flows at a constant rate of 4 L/min into a large ta…

A brine solution of salt flows at a constant rate of 6 L/min into a large ta…

A brine solution of salt flows at a constant rate of 8 L/min into a large ta…

Salt Concentration A tank holds 100 gal of water that contains 20 Ib of diss…

A tank contains 1000 $\mathrm{L}$ of pure water. Brine that contains 0.05 $\…

Decomposition of Salt in Water Salt (NaCl) decomposes in water into sodium (…

A tank contains 1000 L of brine with 15 $\mathrm{kg}$ of dissolved salt. Pur…

A tank contains 1000 L of brine with 15 kg of dissolved salt. Pure water ent…

A $1000$ L tank contains 500 L of water with a salt concentration of 10 $\ma…

03:21

Give the proper trigonometric substitution and find the transformed integral…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.