# Thread: sphere cone problem

1. ## sphere 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.

2. ## Volume of sphere and come

Originally Posted by GAdams
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.
The 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?

3. Originally Posted by GAdams
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.
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:

$\displaystyle \dfrac{a_{cs}}{\pi s^2} = \dfrac{2\pi r}{2\pi s}~\implies~a_{cs} = \dfrac{2\pi r}{2\pi s} \cdot \pi s^2 = \boxed{\pi r s}$

4. Originally Posted by Grandad
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?

Hmm...not sure.

5. ## Surface area of cone

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.

6. Originally Posted by Grandad

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.

s = root 7

Is teh answer 3 pi root7 ?

7. Originally Posted by GAdams
... The radius of the base of a cone is also 3 cm.
...
Originally Posted by Grandad
[SIZE=3]...

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

...
Originally Posted by GAdams
Hmm...not sure.
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:

$\displaystyle a_{cs} = \pi\cdot 3\ cm \cdot 5\ cm = 15 \pi\ cm^2$

8. Originally Posted by earboth
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:

$\displaystyle a_{cs} = \pi\cdot 3\ cm \cdot 5\ cm = 15 \pi\ cm^2$
Ah yes, I made a basic error. I have it now. Thanks!

9. ## Dont understand

Originally Posted by Grandad
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?

Hi i dont understand why you hve multiplied the volume of the cone by 3

10. ## Sphere and cone

Hello osmosis786
Originally Posted by osmosis786
Hi i dont understand why you hve multiplied the volume of the cone by 3

I think you're going to kick yourself!

Originally Posted by GAdams
The radius of a sphere is 3 cm. The radius of the base of a cone is also 3 cm.
Originally Posted by GAdams
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.