solve-the-problems-in-related-rates-the-radius-r-in-mathrmm-of-a-ring-of-a-certain-holograph-an-im-2

Question

Solve the problems in related rates.
The radius $r$ (in $\mathrm{m}$ ) of a ring of a certain holograph (an image produced without using a lens) is given by $r=\sqrt{0.4 \lambda},$ where $\lambda$ is the wavelength of the light being used. If $\lambda$ is changing at the rate of $0.10 	imes 10^{-7} \mathrm{m} / \mathrm{s}$ when $\lambda=6.0 	imes 10^{-7} \mathrm{m},$ find the rate at which $r$ is changing.

Bobby Barnes · Accepted Answer

So there is some white House that is four miles away from the closest point on the beach. Hit the lighthouse rotates clockwise at a constant rate of 10 revolutions permanent. We want to determine how fast the beam of light move across the beach when it&#x27;s two miles away from the closest point. Sorry. So this is a drawing of what we have now. One thing all make Noto is any time we&#x27;re working with, like, miles and revolutions or anything like that, we should convert revolutions to radiance. So converting this to radiance, we would do one rev is equal to to pie. So now the Revolution&#x27;s council out with each other, and we&#x27;d have 20 Hi for a minute for our change in data. All right, now we need to find some way to relate the side we&#x27;re looking for. So let&#x27;s call this side here, E. And we want to find the rate of change of this, which is going to be D B bye gt. So we don&#x27;t know what this is. Now notice if we use tangent of data, this is opposite over adjacent. So this would be be over four and we can go ahead and multiply each side by four to get that B is equal to for Tanja data. And now taking the derivative of this with respect to time on each side we get TB by DT is secretive for and that the derivative of Tanja iss seek it squared data. And then we would have to multiply this by debate about E. T. So we want to figure out what data is or triangle here. Or we can better yet, just figure out what seek it should be in the case of the light being two miles away from that point. Pete, let&#x27;s do that off in the corner here. So the distance between elope was always before and over here it is two for the one we&#x27;re interested in. So now seek it of beta seeker. Dana, this is high pot, a noose over adjacent because it&#x27;s one of our coastline. So we know are a Jason is so Jason Data here would be or and then our hard part news. We confined with Paige risotto two squared must force weird all square root. So then we would have 16 plus fours that the square root of 20 And if we want, we could rewrite this as two times route by by pulling out of four. So you have two times, but and then again, that simplifies down toe five over Or route five over too. But we want to square that. So when we plug this in, we&#x27;d get five over four times our dictated by t t, which was 20. Hi. So 20 pie and well, now notice these pies counts out with each other and then five times 20 would be 100. Hi. And so the rate for this would be in miles per minute.

Cory Kuzinski · Answer

here we go with the illumination. We&#x27;re told illumination varies inversely with the square of the distance from the source. So distance squared. Great. Let&#x27;s figure out K because we&#x27;re told the illumination of a light source is five. Ah, five meters is 70. I don&#x27;t know how to pronounce that. Could Candela? Yeah, that&#x27;s right. So 70 is the elimination. Candela equals k over five squared. So just multiply both sides by 25 because that&#x27;s five squared 70 times 25. Okay, is 1750. All right, So ah, illumination equals 1750 divided by distance squared. And we&#x27;re gonna figure out the illumination. And if we walk a little further away, we&#x27;re 12 meters, Wade, sort of five meters away. So 17 50 divided by 12. Square 17. 50. Divided by 1 44 is what I&#x27;m going to tell you, Benjamin Calculator. And that is what I was around in here. Um, they don&#x27;t tell us around, but we&#x27;re gonna need to, so we&#x27;ll go abouts one doesn&#x27;t place 12.2. Um, kendo, Uh

Amrita Bhasin · Answer

we know the formula for the volume of sphere with a radius are his vehicles 4/3 R cubed times pi or pie are cute differentiating using the chain rule we get Devi over. DT is for our square times Pi di are over dt As you know d r over dt is one r is 20 so view plug and four times 20 squared times pi times one we end up with dif Devi over dt is 1600 py meters cubed per second.

Charles Carter · Answer

Okay, so in this problem, we have a search light on a highway. It&#x27;s gonna rotate. Oh, well, that&#x27;s 100 meters. It&#x27;s gonna rotate through an angle. Fate, uh, we&#x27;re told it&#x27;s rotating at a speed of pi over six radius is per second. Okay, so I&#x27;m gonna change this to B in terms of feta. And so a radius any given time is just gonna be the square root of X squared plus 100 squared And writing that in terms of data would just mean we need to say, Well, co sign of state, uh is equal to 100 divided by the square room of X squared plus out and squared. And so if we want to plug that in for if we want to get that radius, we have to do 100 divided by co sign data eso replacing that. Then we have the speed. Ah, that is traveling is gonna be related to pi over six times 100 divided by cosign data. It&#x27;s really only makes sense for a domain from negative Piper to two pi over two. Because it&#x27;s not your shining on the highway. Okay, so that&#x27;s our speed equation. eso If we take the derivative, we&#x27;ll have 100 pie over six Negative. 100 pi over six Cho Sun Data squared times negative Sign data The directive of our coastline. So from negative pi over to deprive er Teoh co sign square coastline won&#x27;t be negative. Won&#x27;t be zero. I went on to worry too much about that. And so to set the sequel to zero, we just have to know where, um 100 pie Scient. Ada is equal to zero, and so that I would have to equal zero. I would give us the maximum speed. Okay. The ah equation that we had in terms of data for how fast it&#x27;s going is right there. Okay, thank you very much.

Amrita Bhasin · Answer

Okay, we know this appears are angled data. This is four miles. This is D, and the speech over here is ex So tan. Data is ops over Jason, which is acts over four. This means that seek in squared data D theta over. DT is one over four D x over GT. This means that d backs over. DT is for over co sign Squared Data D is the square root of four squared plus one squared, which is the square root of 17 miles. Therefore, co signed data is four over D four, divided by the square of 17 to determine co sign squared data. We square the numerator and denominator, We end up with 16 over 17. Okay, now we&#x27;re gonna be calculating DX over DT. We know that 10 revolutions per minute times two pi revolutions, but times to pay Iranians and one revolution gives us 20 pie radiance per minute. Therefore, we know that d acts over. DT is four or 16 over 17 to have a complex direction. They&#x27;re multiplying times 20 pie, which is the rate of radiance per minute giving us 85 pie miles per minute

Problem

Solve the problems in related rates. An Earth sat…

Problem 15 Easy Difficulty

Answer

Related Courses

Basic Technical Mathematics with Calculus

Related Topics

Discussion

Top Calculus 1 / AB Educators

Grace H.

Caleb E.

Kristen K.

Michael J.

Calculus 1 / AB Courses

Precalculus Review - Intro

Functions on the Real Line - Overview

Recommended Videos

Watch More Solved Questions in Chapter 24

Video Transcript

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Caleb E.

Kristen K.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

Basic Technical Mathematics with Calculus

Grace H.

Caleb E.

Kristen K.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Grace H.

Caleb E.

Kristen K.

Michael J.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus