Problem based learning

**poppy_wolf** · January 16th 2006, 07:08 AM

I am not really sure what type of question is this. The question is below. Thank you.

A hollow hemispherical bowl of radius 4cm has a smaller solid spherical marble of radius 1cm at the bottom. If the bowl is filled with water to a depth of 1.5cm, find the volume of water in the bowl.

**dud** · January 16th 2006, 07:39 AM

Calculate the volume of a hemispherical object with a radius of 1.5cm - the volume of the ball which is submerged in the water?

**ThePerfectHacker** · January 16th 2006, 03:06 PM

Originally Posted by poppy_wolf

I am not really sure what type of question is this. The question is below. Thank you.

A hollow hemispherical bowl of radius 4cm has a smaller solid spherical marble of radius 1cm at the bottom. If the bowl is filled with water to a depth of 1.5cm, find the volume of water in the bowl.

I would solve this problem with Calculus. The most important step is to know that given a hemisphere filled with water to depth of $a$ having radius $r$ then its volume is $\pi (\frac{2}{3}r^3-ar^2+\frac{1}{3}a^3)$ . Then from that your subtract the volume of the marble inside which in this case has volume $\frac{4}{3}\pi$

**poppy_wolf** · January 17th 2006, 08:03 AM

Originally Posted by ThePerfectHacker

I would solve this problem with Calculus. The most important step is to know that given a hemisphere filled with water to depth of $a$ having radius $r$ then its volume is $\pi (\frac{2}{3}r^3-ar^2+\frac{1}{3}a^3)$ . Then from that your subtract the volume of the marble inside which in this case has volume $\frac{4}{3}\pi$

Will the solution be like this..

Volume of bowl = 2/3 (4^3) - (1.5)(4^2) + 1/3 (1.5^3)
= 17 13/24

volume of water = 17 13/24 - $<br /> \frac{4}{3}\pi<br />$

= 13.35287

**ThePerfectHacker** · January 17th 2006, 02:13 PM

Did you forget to multiply by $\pi$ in the first one?

**earboth** · January 18th 2006, 11:42 AM

Hello,

to make sure that I got your question I've attached a sketch of the situation as I understand it.

The marble has a diametre of 2 cm, that means it will stuck out of the water if there is only a depth of 1.5 cm.
Therefore you have to calculate two parts of spheres:

$V_{part\ of\ sphere}=\frac{\pi}{3}\cdot h^2 \cdot (3\cdot r-h)$

$V_{1}=\frac{\pi}{3}\cdot 1.5^2 \cdot (3\cdot 4-1.5)$

$V_{2}=\frac{\pi}{3}\cdot 1.5^2 \cdot (3\cdot 1-1.5)$

The difference of both volumes is the volume of water you're looking for.

Bye

**ThePerfectHacker** · January 18th 2006, 01:34 PM

I see what you are saying earboth. I did not take into account the fact that the sphere stuck out of the water!

This is definetly a Calculus problem for volume. It is solved with the formula I posted here before.

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