The radius of a sphere is 3 cm. The radius of the base of a cone is also 3 cm.

The volume of the sphere is 3 times the volume of the cone.

Work out the curved surface area of the cone, giving your answer as a multiple ofpi.

Thanks.

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- Mar 8th 2009, 09:19 AMGAdamssphere cone problem
The radius of a sphere is 3 cm. The radius of the base of a cone is also 3 cm.

The volume of the sphere is 3 times the volume of the cone.

Work out the curved surface area of the cone, giving your answer as a multiple of*pi*.

Thanks. - Mar 8th 2009, 09:51 AMGrandadVolume of sphere and come
Hello GAdamsThe volume of a sphere is $\displaystyle \frac{4}{3}\pi r^3$, and the volume of a cone is $\displaystyle \frac{1}{3}\pi r^2h$. So:

$\displaystyle \frac{4}{3}\pi r^3 = 3\times \frac{1}{3}\pi r^2h = \pi r^2h$

$\displaystyle \Rightarrow h = \frac{4}{3}r = 4$ cm.

Can you finish it now?

Grandad - Mar 8th 2009, 11:13 AMearboth
Let s denote the slanted height, r the radius of the base circle and h the height of the cone. Then the curved surface of a cone is a sector of a circle with radius s. The area of the curved surface, the area of the complete circle, the circumference of the base circle = arc of the sector and the circumference of the complete circle are directly proportional:

$\displaystyle \dfrac{a_{cs}}{\pi s^2} = \dfrac{2\pi r}{2\pi s}~\implies~a_{cs} = \dfrac{2\pi r}{2\pi s} \cdot \pi s^2 = \boxed{\pi r s}$ - Mar 8th 2009, 11:17 AMGAdams
- Mar 8th 2009, 11:25 AMGrandadSurface area of cone
Hello GAdams

earboth has given you the formula for the curved surface area of a cone. You now need to work out the value of $\displaystyle s$, which you can do using Pythagoras' Theorem. (It's the easiest right-angled triangle of all!)

Then plug the values of $\displaystyle r$ and $\displaystyle s$ into this formula, and you're there.

Grandad - Mar 9th 2009, 12:37 PMGAdams
- Mar 9th 2009, 12:44 PMearboth
Since r = 3 cm and h = 4 cm the slanted height s is:

$\displaystyle s^2 = r^2 + h^2~\implies~ s = 5\ cm$

Therefore the curved surface area is:

$\displaystyle a_{cs} = \pi\cdot 3\ cm \cdot 5\ cm = 15 \pi\ cm^2$ - Mar 9th 2009, 01:09 PMGAdams
- Mar 14th 2009, 04:48 AMosmosis786Dont understand
- Mar 14th 2009, 05:58 AMGrandadSphere and cone
- Mar 14th 2009, 08:11 AMosmosis786Your absolutely correct
i never read the question properly haha(Speechless)