{Use of Tech} Hours of daylight The number of hours of dayli...

Question

{Use of Tech} Hours of daylight The number of hours of daylight at any point on Earth fluctuates throughout the year. In the Northern Hemisphere, the shortest day is on the winter solstice and the longest day is on the summer solstice. At 40° north latitude, the length of a day is approximated by D(t) = 12−3 cos (2π(t+10) / 365), where D is measured in hours and 0≤t≤365 is measured in days, with t=0 corresponding to January 1.
b. Find the rate at which the daylight function changes.

b. Find the rate at which the daylight function changes.

Accepted Answer

Welcome back, everyone. In this problem, a farmer is analyzing the growth pattern of a particular crop, which can be approximated by the function G of M equals 5 minus 2 costs of 2 pi multiplied by M + 15 divided by 365, where G represents the height of the crop in meters and M is the number of days since planting on March 1st, considered as day zero. What is the rate of growth of the crop with respect to this at any given day. A says it's 4 pi divided by 365 multiplied by the cosine of 2 pi multiplied by M + 15 divided by 365. B says it's 4 pi divided by 365 multiplied by the sine of 2 pi multiplied by M + 15 divided by 365. C says it's pi divided by 365 multiplied by the sign of 2 pi multiplied by M + 15 divided by 365. And the D says it's pi divided by 365 multiplied by the cosine of 2 multiplied by M + 15 divided by 365. Now if we're going to find the rate of growth of the crop with respect to days at any given day M, then we'll need to find a derivative of G with respect to M because that is what the rate of growth represents when we're talking here about the rate, OK. So we'll need to differentiate GFM. No, to differentiate GM here, notice that GM has two terms, so we can differentiate our constant 5 and then we can differentiate our cosine term. OK. So no. That means the derivative of G with respect to M is going to be equal to the derivative with respect to m of 5 minus the derivative with respect to M of 2 multiplied by the cosine of 2 multiplied by M + 15 divided by 365. OK. So now how can we do that? Well, when we differentiate a constant, it equals 0. So this, the derivative of 5 with respect to M is going to be 0, OK. Next, to differentiate the cosine term, we know the derivative of the cosine is sine. So to differentiate this term, we'll differentiate the cosine which is sign and then multiply that by the derivative of what's inside the cosine argument based on the chain rule. So no, the derivative of cosine is negative sign, sorry, not negative sign. So that's gonna be -2, multiplied by the sign of 2 pi M + 15 divided by 365. OK. And now we're going to multiply that by the derivative of 2 multiplied by M + 15 divided by 365, which is going to be equal to 2 to 2 divided by 365, OK, because here. When we differentiate M, it equals 1, differentiate 15, it equals 0, OK? So that's 2 pi divided by 365. That's what we would end up with. Now if we expand our our idea here, OK. If we're subtracting a negative, it's going to become positive, and we can multiply 2 by 2 pi to get 4 pi. So that's 4 divided by 365 multiplied by the sine of 2 pi multiplied by M+ 15 divided by 365. This is the derivative of G with respect to m. If we look back on our answer choices, that tells us then that B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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