Solving Related Rates Prob: Check Work w/ Hemispherical Bowl

In summary, a hemispherical bowl of radius 8 in. is being filled with water at a constant rate. If the water level is rising at the rate of 1/3 in./s at the instant when the water is 6 in. deep, find how fast the water is flowing in by using the fact that if V is the volume of the water at time t, then dV/dt = (pi)(r)^2(dh/dt) The first formula you show looks fine, but can you tell us how you arrived at your second formula, namely dV/dt = pi (64 - (8 - h)^2) dh/dt?
  • #1
elitespart
95
0
Related rates prob. [solved]

A hemispherical bowl of radius 8 in. is being filled with water at a constant rate. If the water level is rising at the rate of 1/3 in./s at the instant when the water is 6 in. deep, find how fast the water is flowing in by using the fact that if V is the volume of the water at time t, then dV/dt = (pi)(r)^2(dh/dt)

I just need someone to check my work. From my picture, I have:

dV/dt = (pi)(64-(8-h)^2) dh/dt.

just plug in 6 for h and 1/3 for dh/dt and solve right? Thanks.
 
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  • #2
The first formula you show looks fine, but can you tell us how you arrived at your second formula, namely dV/dt = pi (64 - (8 - h)^2) dh/dt?
You mentioned that it comes from your picture, but since I can't see your picture, it's not clear to me how h is related to the radius of the disk you are using for your incremental volume. I'm not saying your formula is wrong, but you haven't provided enough information so that I can tell.
 
  • #3
Mark44 said:
The first formula you show looks fine, but can you tell us how you arrived at your second formula, namely dV/dt = pi (64 - (8 - h)^2) dh/dt?
You mentioned that it comes from your picture, but since I can't see your picture, it's not clear to me how h is related to the radius of the disk you are using for your incremental volume. I'm not saying your formula is wrong, but you haven't provided enough information so that I can tell.

http://img363.imageshack.us/img363/3018/problemab0.jpg http://g.imageshack.us/img363/problemab0.jpg/1/

I suck at drawing on paint but this is kind of how my pic looks. y + h is radius of the bowl which is 8. 8 - h = y and then i just used c^2 - a^2 = b^2. And thanks for your help.
 
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  • #4
elitespart said:
http://img363.imageshack.us/img363/3018/problemab0.jpg http://g.imageshack.us/img363/problemab0.jpg/1/

I suck at drawing on paint but this is kind of how my pic looks. y + h is radius of the bowl which is 8. 8 - h = y and then i just used c^2 - a^2 = b^2. And thanks for your help.


OK, thanks for including the drawing. Now I know how h and y are related. I don't see anything wrong in your second formula.
 
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  • #5
Thanks again.
 

FAQ: Solving Related Rates Prob: Check Work w/ Hemispherical Bowl

How do I set up a related rates problem involving a hemispherical bowl?

To set up a related rates problem with a hemispherical bowl, you must first identify the variables involved and their relationships. This usually involves using the formula for the volume of a sphere (V=4/3πr³) and differentiating it with respect to time. Then, you can substitute in the given values and solve for the unknown rate of change.

What is the key to solving related rates problems with a hemispherical bowl?

The key to solving related rates problems involving a hemispherical bowl is to recognize the relationship between the changing variables. In most cases, this will involve using the Pythagorean theorem to relate the radius and height of the bowl. It is important to carefully label and draw a diagram of the situation to help visualize the problem.

How do I know if my solution to a related rates problem with a hemispherical bowl is correct?

One way to check if your solution is correct is to plug the values back into the original equations and see if they satisfy the given conditions. Another way is to compare your solution with other known values or use common sense to determine if the result makes sense in the given context.

Can I use calculus to solve any related rates problem with a hemispherical bowl?

Yes, calculus can be used to solve any related rates problem involving a hemispherical bowl. However, it is important to first understand the problem and the relationships between the changing variables before applying calculus techniques. Without a solid understanding of the problem, it may be difficult to set up the equations correctly.

Do I need to use a specific formula or method to solve related rates problems with a hemispherical bowl?

No, there is not one specific formula or method for solving related rates problems with a hemispherical bowl. Each problem may require a different approach depending on the given information and relationships between the variables. It is important to carefully analyze the problem and determine the most efficient way to solve it.

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