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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 $0.10 \times 10^{-7} \mathrm{m} / \mathrm{s}$ when $\lambda=6.0 \times 10^{-7} \mathrm{m},$ find the rate at which $r$ is changing.

Calculus 1 / AB

Chapter 24

Applications of the Derivative

Section 4

Related Rates

Derivatives

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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'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'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's council out with each other, and we'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're looking for. So let'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'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'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're interested in. So now seek it of beta seeker. Dana, this is high pot, a noose over adjacent because it'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'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.

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