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Stiles-Crawford effect Light enters the eye through the pupil and strikes the retina, where photoreceptor cells sense light and color. W. Stanley Stiles and B. H. Crawford studied the phenomenon in which measured brightness decreases as light enters farther from the center of the pupil. (See the figure.) They detailed their findings of this phenomenon, known as the Stiles-Crawford effect of the first kind, in a paper published in $1933 .$ In particular, they observed that the amount of luminance sensed was not proportional to the area of the pupil. The percentage $P$ of the luminance entering a pupil at a distance of $r \mathrm{mm}$ from the center that issensed at the retina can be described by$$P=\frac{100\left(1-10^{-\rho r^{2}}\right)}{\rho r^{2} \ln 10}$$where $\rho$ is an experimentally determined constant, typically about $0.05 .$(a) If light enters a pupil at a distance of 3 $\mathrm{mm}$ from the center, what is the percentage of luminance that is sensed at the retina?(b) Compute the percentage of luminance sensed if light enters the pupil at a distance of 2 $\mathrm{mm}$ from the center. Does it make sense that it is larger than the answer topart (a)?(c) Compute lim $_{r \rightarrow 0} P .$ Is the result what you would expect?

a. 62.3$\%$b. 80.1$\%$c. 100$\%$

Calculus 1 / AB

Chapter 4

Applications of Derivatives

Section 3

L'Hospital's Rule: Comparing Rates of Growth

Differentiation

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Okay, So for this problem, we are told that the percentage of blue minutes is going to be equal to 100 times one minus 10 to the power of negative ro. Times are square over road times are squared natural love of 10. So part ay asks us if light enters the people at a distance of three millimeters, which will be our are the distance. What is the percentage wasn't a plug. We're plugging our equals three in and we were given row is equal to 0.5 So if we plug that in to Pee wee get 100 times one minus 10 to the power of negative 0.5 times three squared over 0.5 times three squared times the natural log of 10. So let's plug this into the calculator and it comes out to being approximately 62.3%. It's a part B is asking us to do the same calculation, but where are is equal to two instead of three. So for that one, we get that P is equal to 100 times one minus 10 to the power of negative role, which is 0.5 That doesn't change. Times two squared over ro 0.5 times, two squared times, the natural log of 10. So when we put this in the calculator, it comes out to approximately 80.1%. So then it asks us, what does it make sense? That R equals two is larger than our equals three. And it does make sense. And this is due to the styles Crawford. If fact that was described. And this effect states that the closer the light entering is the brighter it is. Okay. And so in this case, the light is closer. So does make sense. That that we would have more greater percentage here and then sees really where the trick comes in. Now it wants us to calculate the limit as our approaches zero of 100. And so this is, of course, our percentage formula. So 100 times 10 to the negative ro r squared and you could put your row in as a 0.5 if you want. Doesn't really make a difference. Rue R squared. Natural out of 10. Okay, well, so we have some Constance here. The 100 the row can come out. So this is equal to and actually soak in the natural log of tents will have 100 over road times Theme natural. Other 10 pulled out times the limit as our approaches zero of one minus 10 to the negative ro r squared over r squared. And if we evaluate this limit, our equal zero we get one minus 10 on top over Joe squared so we can actually use low petals rule on this fraction. So this is going to be equal to 100 over road times, the natural out of 10 times the limit as our approach to zero. And now we're gonna take the derivative of the top which is going to become this one a zero and then this derivative of the negative 10 to the power of negative road, Times Square is going to be the natural log of 10 times five to the negative ro r squared times two to the negative ro R squared plus one times are times row. Okay. And then the derivative are square will be to our We have a few things happening here. Thes ours will cancel out this too is actually gonna cancel out this. Plus one in the exponents here and then the row and the natural order of 10 we can pull out, right? And those air going to then cancel with this row and natural of 10 that we've already pulled out. So when we rewrite this, we get 100 times the limit as our approaches zero. And now that these have the same exponents, they could be combined 10 to the negative Rue R squared. Yes, and I we put zero in for our if we evaluate it and we're going to end up with 100 times 10 to 0, which is one is equal to 100. Okay, so in this case, we're calculating percentage, so it's 100%.

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