Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

The answers to most of the following exercises are in terms of logarithms and exponentials. A calculator can be helpful, enabling you to express the answers in decimal form.The intensity $L(x)$ of light $x$ feet beneath the surface of the ocean satisfies the differential equation$$\frac{d L}{d x}=-k L$$.As a diver, you know from experience that diving to $18 \mathrm{ft}$ in the Caribbean Sea cuts the intensity in half. You cannot work without artificial light when the intensity falls below one-tenth of the surface value. About how deep can you expect to work without artificial light?

$$59.8 \mathrm{ft}$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

Integrals and Transcendental Functions

Section 2

Exponential Change and Separable Differential Equations

Functions

Trig Integrals

Missouri State University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

02:24

Working underwater The int…

04:10

The answers to most of the…

10:12

02:11

04:53

03:13

03:23

03:25

03:12

02:31

So we have this differential equation that describes how the intensity of light changes as we are ex feet below the surface of the ocean. We're told that 18 feet below the surface would give us half of our intensity that we would get from the surface. And we want to determine at what point will must diving down reach 1/10 the value of the surface level. But I have 1/10 of that shack Should be 1/10 l not right. So it doesn't matter the l knots you'll see as we go through the steps so cancel out, but just kind of writing get out the way they, uh, state and maybe should also write that l of zero is l'm not. That will be our initial surface. So let's go ahead and start solving this differential equations. So first, we're gonna divide each side by l. So we're gonna get DEA one over L d O is equal to negative K DX. If we were toe, multiply the DX over. Now, Once we have this different this incontestable equation, we could go ahead and integrate each side. The left hand side should integrate to the natural log of the absolute value of, though is equal to then on the right hand side. Negative K X, plus some constancy. No, Since they don't want us to solve for the differential equation, all they want us to do is figure out. When does this hold? We need to start plugging everything into here. Don't stoop to that. So let's use our first condition. Oh, that l of zero is zero to El Ni so we can plug that into give so l of zero Oh, so the absolute value of el not will intensity should be positive. So absolute value. That should just be l'm not so we don't have anything to worry about that if we just dropped the absolute value and then this is going to be equal to negative k time zero, which is gonna be zero plus our constancy. So we just solved to see that C is equal to the natural log of the absolute or the natural log of l not. All right, so that's seems pretty good so far. Now that shoes are second condition here, help us all. For what K is so remember. This is saying X is equal to 18 and l is gonna want half l not. So we're gonna have a natural log of 1/2. No, not is equal to the negative of K times 18 plus the natural log of L. Not now. We want to get Kay by itself. So we're gonna want to subtract hell. Not over. Er the natural log a bell knots. That's gonna give us natural log of 1/2 l not minus natural. Log of El Not. And then in doing that, we would just be left with K times. Negative eight teams Such divide this by negative 18 while we're at it. And this should give us our value for Kay. Now, notice when we have to natural logs being subtracted like this, we can write it as the division of the insides. So that would actually give us that. Kay is even too negative. Won over 18 of the natural log of 1/2. All right, so we have this snow and let's get rid of that negative outfront. Remember, we can do that by pulling it in to be the power. Oh, 1/2 of the natural log by the power rule. So there's gonna be 1/18 natural log of two. Hurry, Majola line a little bit better now let's just plug everything in and write out our equation. This start. So we have that Ellen of the natural log of L. There's going to be equal to negative Ex Natural log up to you over 18 plus the natural log. Oh, l not. So this is the equation that we have. Well, let's go ahead and plug in to figure out when this should be good to 1/10 of l not. So we want to solve for X this time. So we're gonna have Ellen is equal to 1/10. Well, not is equal to negative X Natural. Log on to over 18 plus natural log of el Not so we're going to subtract over. So, just like before, the natural log should combine into one. And the l knots would just cancel out, which would give us Ellen of 1/10. So this is why I was saying at the start, the natural logs are the l knots Don't really matter all that much. So we have negative X natural log. 02 all over 18. Now just solving for X we should end up with X is equal to And I'm gonna rewrite this as negative natural log of 10. So we're going to multiply by 18 and then divide by negative natural log to sew these negatives here counts out with each other. And then So this would be our exact answer and feet bullets actually approximate this because I have no idea what that number is even supposed to mean. So natural Aga 10 is about 2.3. Multiply that by 18 get about 41 then divide that by the natural log of two. And this gives about 59 27 9 beat. So around this here is wind. We would need to use some kind of artificial life.

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…

Working underwater The intensity $L(x)$ of light $x$ feet beneath the surfac…

The answers to most of the following exercises are in terms of logarithms an…

At what points are the functions continuous?$$y=\tan \frac{\pi x}{2}$$

10:25

We say $f$ is uniformly continuous on $[a, b]$ if, given any $\varepsilon &g…

03:05

If you average $30 \mathrm{mi} / \mathrm{h}$ on a $150-\mathrm{mi}$ trip and…

01:59

Find the derivatives.a. by evaluating the integral and differentiating t…

03:11

Evaluate the integrals.$$\int \operatorname{coth} \frac{\theta}{\sqrt{3}…

02:15

Evaluate the integrals using integration by parts.$$\int t^{2} e^{4 t} d…

03:02

Evaluate the integrals. Some integrals do not require integration by parts.<…

05:06

Use a CAS to perform the following steps:a. Plot the functions over the …

17:47

Use the method of Example 4a or Equation (1) to evaluate the definite integr…

02:27

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.