Question 1 water fills a conical cup such that the the...

Question 1. Water fills a conical cup such that the the height of the water in the cup is constantly rising at a rate of 3 cm/s. The cup itself has a radius of 10 cm and a height of 20 cm. a) Let r be the radius of the surface of the water and h be the height. What is dr/dt when h = 12? b) Let V be the volume of water. What is dV/dt at the same moment in time?

Transcript

00:01 Hi there, so for this problem we have a situation something like this, okay, so we have a spherical bowl and the radius that we are given for this bowl, about here, that radius will be equal to 10 centimeters, right, and the depth of the water in the bowl we're gonna leave it as h, okay, so let's put that, we have some water in this bowl, something like this, all right, and then the height of this water is about here, so we will have a different radius in here, we're gonna label this the radius r, okay, this is the height h about here, now we're given some volume expression for this, let's see, that will be that the volume is pi times the height square, then this divided by 3 and then this times 30 minus h, once we have this, then we're told the water is pouring into the bowl at a rate, so the rate change of the volume with respect to time is equal to 2 cubic centimeters per second, now for part a of this problem we are told that if the radius of the water surface is r, state r in terms of h, so to state r in terms of h we must remember that there is a relationship for this, we know that in our case we can obtain this by trigonometry, so we know that this difference in here, we know that this will be 10, right, and then this difference in here will be then 10 minus h, right, and then well this remaining h, right, so we know then that this will be the radius to the square plus, then we will have that this is 10 minus h and that to the square, then this equals to the radius that we're given, 10 and that to the square, so let's solve for the radius square, so that will be then 10 to the square minus 10 minus h to the square, so let's simplify this in here, ok, so when we do this we obtain a value of, so the radius is just 20 times h minus h square, so if we want to solve, well this is the radius square, so if we want to solve for the radius that will be the square root of 20 times h minus h square, then we substitute that into the volume, so the volume is equal to pi times, well, sorry, we just leave that just like this because that is the question for part a of this problem, the question for part a was to, if the radius of r is a centimeter, so this r in terms of h, so that's what we did in here, now for part b of this problem the question is when h is equal to 4, we need to determine the radius of change of indihide and the radius of the water surface, so first we need to derivate this with respect to the height, so with…

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