Find the volume of a sphere with respect to time

In summary, the conversation revolves around a problem involving the volume of an object with a known volume of 100 cm. The person is struggling to understand why the chain rule requires them to take the inverse of 4∏r^2, but eventually realizes that it is a matter of solving for \displaystyle \frac{dr}{dt} in the equation \displaystyle \frac{dV}{dt}=4\pi r^2\frac{dr}{dt}\,.
  • #1
bobsmith76
336
0

Homework Statement



Screenshot2012-01-27at25235PM.png


I guess I should point out that the volume was known, which is 100 cm.

The Attempt at a Solution



I don't see why using the chain rule compels you to take the inverse of 4∏r^2. Any help would be appreciated.

I have a feeling that the solution is something like

100 = (4pir^2)/x

but that still is not right.
 
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  • #2
bobsmith76 said:
Screenshot2012-01-27at25235PM.png


I don't see why using the chain rule compels you to take the inverse of 4∏r^2. Any help would be appreciated.

They're simply solving [itex]\displaystyle \frac{dV}{dt}=4\pi r^2\frac{dr}{dt}\ \text{ for }\ \frac{dr}{dt}\,.[/itex]
 
  • #3
Ok, now I got it. The way the book proceeded was a bit bewildering. I didn't see how dV/dt = 100, and r = 25, but now I do.
 

1. How do you calculate the volume of a sphere with respect to time?

The formula for finding the volume of a sphere with respect to time is V(t) = (4/3)πr^3, where V(t) represents the volume at a particular time and r is the radius of the sphere.

2. What is the unit of measurement for the volume of a sphere with respect to time?

The unit of measurement for the volume of a sphere with respect to time is typically cubic units, such as cubic meters or cubic feet.

3. Can the volume of a sphere change over time?

Yes, the volume of a sphere can change over time if the radius of the sphere changes. As the radius increases, the volume also increases, and as the radius decreases, the volume decreases.

4. What is the relationship between the volume of a sphere and its radius over time?

The volume of a sphere is directly proportional to the cube of its radius. This means that as the radius increases or decreases, the volume will change by a factor of that change cubed.

5. Can the volume of a sphere with respect to time be negative?

No, the volume of a sphere with respect to time cannot be negative. Volume is a measure of space, and it cannot have a negative value. If the formula for finding the volume results in a negative value, it means that the radius was entered incorrectly or the calculation was done incorrectly.

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