Results 1 to 11 of 11

Math Help - sphere cone problem

  1. #1
    Member GAdams's Avatar
    Joined
    Apr 2007
    Posts
    171
    Awards
    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1

    Volume of sphere and come

    Hello GAdams
    Quote Originally Posted by GAdams View Post
    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 \frac{4}{3}\pi r^3, and the volume of a cone is \frac{1}{3}\pi r^2h. So:

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

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

    Can you finish it now?

    Grandad
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,830
    Thanks
    123
    Quote Originally Posted by GAdams View Post
    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:

    \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}
    Attached Thumbnails Attached Thumbnails sphere cone problem-kegelmantel.png  
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member GAdams's Avatar
    Joined
    Apr 2007
    Posts
    171
    Awards
    1
    Quote Originally Posted by Grandad View Post
    Hello GAdamsThe volume of a sphere is \frac{4}{3}\pi r^3, and the volume of a cone is \frac{1}{3}\pi r^2h. So:

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

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

    Can you finish it now?

    Grandad
    Hmm...not sure.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1

    Surface 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 s, which you can do using Pythagoras' Theorem. (It's the easiest right-angled triangle of all!)

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

    Grandad
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member GAdams's Avatar
    Joined
    Apr 2007
    Posts
    171
    Awards
    1
    Quote Originally Posted by Grandad View Post
    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 s, which you can do using Pythagoras' Theorem. (It's the easiest right-angled triangle of all!)

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

    Grandad
    s = root 7

    Is teh answer 3 pi root7 ?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,830
    Thanks
    123
    Quote Originally Posted by GAdams View Post
    ... The radius of the base of a cone is also 3 cm.
    ...
    Quote Originally Posted by Grandad View Post
    [SIZE=3]...

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

    ...
    Quote Originally Posted by GAdams View Post
    Hmm...not sure.
    Since r = 3 cm and h = 4 cm the slanted height s is:

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

    Therefore the curved surface area is:

    a_{cs} = \pi\cdot 3\ cm \cdot 5\ cm = 15 \pi\ cm^2
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member GAdams's Avatar
    Joined
    Apr 2007
    Posts
    171
    Awards
    1
    Quote Originally Posted by earboth View Post
    Since r = 3 cm and h = 4 cm the slanted height s is:

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

    Therefore the curved surface area is:

    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!
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Mar 2009
    Posts
    28

    Dont understand

    Quote Originally Posted by Grandad View Post
    Hello GAdamsThe volume of a sphere is \frac{4}{3}\pi r^3, and the volume of a cone is \frac{1}{3}\pi r^2h. So:

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

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

    Can you finish it now?

    Grandad

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

    thanks ..please reply asap
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1

    Sphere and cone

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

    thanks ..please reply asap
    I think you're going to kick yourself!

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

    Grandad


    Follow Math Help Forum on Facebook and Google+

  11. #11
    Junior Member
    Joined
    Mar 2009
    Posts
    28

    Smile Your absolutely correct

    i never read the question properly haha
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Maximisation Problem - Cone/Sphere
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 6th 2011, 03:22 PM
  2. sphere cone problem
    Posted in the Geometry Forum
    Replies: 1
    Last Post: January 31st 2010, 10:18 AM
  3. sphere and cone
    Posted in the Geometry Forum
    Replies: 1
    Last Post: December 22nd 2008, 08:59 PM
  4. Sphere inscribed in cone problem
    Posted in the Calculus Forum
    Replies: 5
    Last Post: December 4th 2008, 06:10 AM
  5. Sphere inside a cone problem.
    Posted in the Geometry Forum
    Replies: 2
    Last Post: July 7th 2007, 11:12 AM

Search Tags


/mathhelpforum @mathhelpforum