What is the percentage of water in a sphere given its volume and radius?

[LiHJ]

Homework Statement

Dear Mentors and PF helpers,

I can do part (a) and (b) but don't really know how to do (c) and (d). Can somebody teach me how to go about solving it.

Homework Equations

Volume of cone: $$\frac{1}{3}πr^2h$$
Volume of cylinder: $$πr^2h$$
Volume of sphere: $$\frac{4}{3}πr^3$$
Curved surface area of cylinder: $$2πrh$$
Curved surface area of cone: $$πrl$$

The Attempt at a Solution

(a) $$π⋅4^2⋅8+\frac{1}{3}π⋅4^2⋅6=502.65≈503cm^3$$
(b)$$2⋅π⋅4⋅8+π⋅4⋅\sqrt{4^2+6^2}=291.67≈292cm^2$$

 

[Simon Bridge]

You can do (a) so you know the volume of the water that gets poured into the sphere - you also know an equation for the volume of a sphere.
Do you also have an equation for the volume of the spherical cap?

 

[LiHJ]

I haven't learned about that, the formula that I listed is what I know from now. Thanks

 

[LiHJ]

I also know about the surface area of a sphere is $$4πr^2$$

 

[Simon Bridge]

I haven't learned about that, the formula that I listed is what I know from now.

... but no matter, you can look it up.

 

[Mentallic]

You don't need the formula for the cap, the information is sufficient to find the answer. Draw a 2d representation of the sphere (a circle) and from the centre of the circle, draw a line directly up connecting it and the water surface and label this length h. Then create a right triangle by connecting the centre and where the water surface touches the circle. This is the radius r.
Now you have a right triangle with unknowns r and h, and by Pythagoras, you have a relationship between those two unknowns. Can you find any other equation that also gives a relationship between r and h? Hint: What information has been given to you that you haven't used yet?

Also, it should be pretty clear that the surface area formulas don't play a role in this question. Don't get distracted by those and just stick to volume.

 

[LiHJ]

 

[LiHJ]

Thank you Mr Bridge and Mentallic for attending to my query

 

[Ray Vickson]

You don't need the formula for the cap, the information is sufficient to find the answer. Draw a 2d representation of the sphere (a circle) and from the centre of the circle, draw a line directly up connecting it and the water surface and label this length h. Then create a right triangle by connecting the centre and where the water surface touches the circle. This is the radius r.
Now you have a right triangle with unknowns r and h, and by Pythagoras, you have a relationship between those two unknowns. Can you find any other equation that also gives a relationship between r and h? Hint: What information has been given to you that you haven't used yet?

Also, it should be pretty clear that the surface area formulas don't play a role in this question. Don't get distracted by those and just stick to volume.

I must be missing something: I don't see how a volume-related question can ignore the volume of the cap---or, rather, the formula for the volume.

 

[Mentallic]

I must be missing something: I don't see how a volume-related question can ignore the volume of the cap---or, rather, the formula for the volume.

You agree that the radius of the sphere can be determined without any volume formulae, correct? Lihu posted the solution.

Well, for question (d), we already know the volume of the water that was poured into the sphere, and we know the radius of the sphere, hence the volume, thus we can find the percentage of water that filled the sphere or even the volume of the cap if we were asked for it.

 

[Simon Bridge]

I must be missing something: I don't see how a volume-related question can ignore the volume of the cap---or, rather, the formula for the volume.

It's a good catch by Mentallic - (c) is not, in fact, a volume-related question. The volume doesn't matter for the answer. The information about the volume of water that fills the sphere to height 10cm is a red herring.

I may be interesting to check that the dimensions given describe the correct volume though.

 

[Ray Vickson]

You agree that the radius of the sphere can be determined without any volume formulae, correct? Lihu posted the solution.

Well, for question (d), we already know the volume of the water that was poured into the sphere, and we know the radius of the sphere, hence the volume, thus we can find the percentage of water that filled the sphere or even the volume of the cap if we were asked for it.

Ahhh... I missed that first, crucial, sentence about where the water came from.