| 
  • If you are citizen of an European Union member nation, you may not use this service unless you are at least 16 years old.

  • You already know Dokkio is an AI-powered assistant to organize & manage your digital files & messages. Very soon, Dokkio will support Outlook as well as One Drive. Check it out today!

View
 

Question 9

This version was saved 14 years, 11 months ago View current version     Page history
Saved by Kristina
on April 14, 2009 at 7:59:11 pm
 

A Calculator MAY NOT be used to help answer this question.

 

Water is draining at the rate of 48π ft3/minute from the vertex at the bottom of a conical tank whose diameter at its base is 40 feet and whose height is 60 feet.

 

(a) Find an expression for the volume of water in the tank in terms of its radius at the surface of the water.

 

(b) At what rate is the radius of the water in the tank shrinking when the radius is 16 feet?

 

(c) How fast is the height of the water in the tank dropping at the instant that the radius is 16 feet?

 

Solution

 

Before you start onto the actual questions, I'd suggest drawing a diagram to get a better understanding of what is occurring. This is especially helpful for visual learners.

 

  

Now let's begin...

 

a) Since it is asking for the volume in terms of the radius, that would mean that the only variable that will be in our equation will be r. The trick to this though is that its asking in terms of the radius with respect to the water's surface. Due to this, we will need to do some similar triangles, because as shown in the diagram, we do not know the exact values for h and r. Let us write down what we have so far.

 

 

As you can see, I have solved for h using similar triangles because as stated earlier, the question is specifically asking for the volume to be in terms of the radius, r. If it had been asked to be in terms of the height, then I would have solved for r instead, leaving just the variable, h, in the equation.

 

Now knowing our value for the height, we can now substitute it into the equation for the volume of a cone. Finishing off the question, we should now be left with...

 

And there you have it, the expression for the volume of the water in the tank in terms of its radius at the surface of the water. Magnifique!

 

 

(b) Now we get to the actual related rates part of the question. First off, we should list what we are given/know and what we are looking for...

 

As you can see, we will be needing the equation for volume that we found in part a. We were also given the rate that the volume is changing with respect to time (in minutes). The rate that is unknown to us, or in other words what we are looking for, is what we are trying to solve for in this question. The Leibniz notation standing for the change in the radius with respect to time, which is exactly what the question is asking for.

 

Now, the first step we should take is to write out the volume in terms of Leibniz notation.

 

To be continued... I'm tired and am starting to get a headache and I want some banana tofu.

Comments (0)

You don't have permission to comment on this page.