Question An inverted conical tank full of water has a...

Question An inverted conical tank full of water has a height of 12 feet and a base radius of 5 feet. How much work is required to pump the water over the upper edge of the tank until the height of the water remaining in the tank is 7 feet? Round to the nearest integer. Note: The weight-density of water is 62.4 lb/ft^3. Provide your answer below: ft-lb

Transcript

00:01 Hi there.

00:02 So for this problem we have the situation that is shown in here.

00:06 We have an inverted conical tank full of water that has height of 12 feet and a base radius of 5.

00:21 And the question is how much work is required to pump the water over the upper edge of the tank until the height of the water remaining is equal to 7.

00:34 So until this is equal to 7.

00:37 Right.

00:38 So we need to obtain that work done for this.

00:40 Okay.

00:41 So first we consider a relationship between the radius and the height.

00:48 Okay.

00:49 So then we can divide the radius that we're given for this that is 5 by the height that is 12.

00:57 So this is the ratio between the radius and the height.

01:01 So in here we can solve for the radius.

01:04 So then the radius.

01:06 Oh well.

01:07 So we can solve for the height.

01:08 Sorry.

01:09 So then the height is equal to.

01:21 Well let's solve for the radius.

01:23 Sorry.

01:24 First let's solve for the radius.

01:25 So that will give us 5 divided by 12 times the height.

01:29 So that's a relationship that we have for the radius.

01:33 Now let's consider a volume of a slice of this liquid about here for example.

01:41 Okay.

01:41 So then this is going to have a height delta in the height and some radius at that point.

01:49 So the volume of that slice, let's label that as ps, is just pi times the radius squared at that part and then this times delta of the height.

02:01 Now in here we can substitute the condition that we obtained for the radius.

02:04 So that will be 5 divided by 12 times the radius and that to the square this delta in the height.

02:12 So now let's simplify this.

02:15 So this will be 25 times pi divided by 144.

02:22 This times the height.

02:28 Oh sorry.

02:28 In here is the height and not the radius.

02:31 Then the height to the square times that differential in the height.

02:37 Now the mass of this slide, the mass will be then the density for this that we are given for this.

02:46 That is 62 .4.

02:52 This times this volume that we are given that will give us the mass for this.

02:59 So this times 25, sorry, 25 times pi divided by 144 times the height squared times the differential in the height...

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