Related Rates - Frustum of a Cone

In summary, the problem involves a large container in the shape of a frustum of a cone being filled with water at a constant rate of 4.2 cubic meters per minute. The question is asking for the rate at which the water level is rising when the water is 1 meter deep. To solve this, the relationship between the height and radius of the cone must be determined, which can be done by drawing a trapezoid on a coordinate system. Once this relationship is established, the volume equation for a frustum of a cone can be used to find the rate of change of the height. Drawing a picture and using a coordinate system is recommended for solving this problem.
  • #1
Temp0
79
0

Homework Statement



A large container has the shape of a frustum of a cone with top radius 9 metres , bottom radius 2 metres , and height 7 metres. The container is being filled with water at the constant rate of 4.2 cubic meters per minute.
At what rate is the level of water rising at the instant the water is 1 metre deep?

Homework Equations


V = 1/3∏ (r^2) * h
Internet Says the volume of a frustum of a cone is V = (∏ * h / 3) (R^2 + Rr + r^2)

The Attempt at a Solution


I don't know where to start with this one, I think I have to find some relationship between the height of the cone and the radius of the cone, however, I don't know which radius to use. Once I isolate h with V, I can just derive it and substitute 4.2 for dV/dt, 1 for h, and find dh/dt. Any tips on how to relate radius to height in this situation? Thanks in advance.
 
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  • #2
Have you drawn a picture?
 
  • #3
Looked at from the side, a frustrum of a cone is a trapezoid. If you set up a coordinate system with origin at the center of the smaller face, then the two sides are two lines, one passing through (r, 0) and (R, h). you can write the equation for that line and so get a relation between x (the radius) and y (the height) at each point.
 
  • #4
Hmm, I've tried drawing a picture, but it didn't help.

That trapezoid idea might work, i'll try it, thanks!
 
  • #5
Temp0 said:
Hmm, I've tried drawing a picture, but it didn't help.
A picture by itself probably won't help, but if you draw it on a coordinate system and identify the points on the picture by their coordinates, as HallsOfIvy suggests, it will be useful.

For problems like these it's always a good idea to draw a picture, as described above.
Temp0 said:
That trapezoid idea might work, i'll try it, thanks!
 
Last edited:
  • #6
Temp0 said:
That trapezoid idea might work, i'll try it, thanks!
This is the picture I was wondering if you drew.
 
  • #7
No picture.
 
  • #8
I meant I was wondering if he had drawn the trapezoid picture Halls had described. I didn't attach a picture.
 

Related to Related Rates - Frustum of a Cone

What is a frustum of a cone?

A frustum of a cone is a geometric shape that is formed when the top of a cone is cut off by a plane parallel to its base. It resembles a cone with a smaller cone or pyramid removed from the top.

What are related rates?

Related rates are mathematical problems that involve finding the rate of change of one quantity with respect to another quantity. In the context of a frustum of a cone, related rates can involve finding the rate of change of the volume, surface area, or other dimensions of the frustum.

What are some real-life applications of related rates with frustums of cones?

Related rates with frustums of cones can be used in various real-world scenarios, such as determining the rate at which water is being drained from a conical tank, finding the rate at which a sand pile is growing, or calculating the rate at which a hot air balloon is rising.

What are some strategies for solving related rates problems with frustums of cones?

One strategy for solving related rates problems with frustums of cones is to draw a diagram and label all given and unknown quantities. Then, use the properties of similar triangles to set up a proportion and find the relationship between the changing quantities. Finally, use implicit differentiation and the chain rule to solve for the desired rate of change.

Are there any common mistakes to avoid when solving related rates problems with frustums of cones?

Yes, some common mistakes to avoid include not clearly defining the given and unknown quantities, forgetting to use the properties of similar triangles, and incorrectly using the power rule for implicit differentiation. It is also important to pay attention to units and make sure they are consistent throughout the problem.

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