I got "unlost" .. Mr B you are partially correct

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Posted by Subhotosh Khan on October 09, 2002 at 11:21:17:

In Reply to: Mr B... you lost me ...I'll do the full problem below posted by Subhotosh Khan on October 09, 2002 at 10:22:54:

: : : : I understand what needs to be done, Mr K, but can't finish it.
: : : : Too many things I haven't seen before.
: : : : :mAOB = T
: : : : I understand "AOB", but what is T? degrees, like 75 degrees? Yes
: : : : What does "m" mean? It is short of "measure of" or "value of" angle AOB
: : : : :From that you can get the area of the base(A_1)
: : : : Why do I need to get the area? Look below
: : : : :volume of small cone (V_1)=1/3*R*A
: : : : where does that come from? V = (pi r^2 h) / 3, is it not? In this case A = pi * r^2 .... and h = R (the radius of the circle you started with) so we are talking about the same thing
: : : ***********************************
: : : Assume mAOB = T (in radians)
: : : So chord AB = R * T (definition of angle)
: : : So the circumference of the base of the small cone
: : : 2* pi * r = R * T ? (r = radius of the circular base of the small cone)
: : : r = R * T/(2*pi)
: : : V_1 = (pi)* r^2 * h / 3
: : : = (pi)* [R * T/(2*pi)]^2 * R / 3
: : : = R^3/(12*pi) * T^2
: : : Similarly find V_2

: : was fooling around with this; interim question, Mr K (I'm rusty with cones!):

: : height of smaller cone = Rsqrt(360^2 - T^2) / 360

: : height of bigger cone = Rsqrt[T(720 - T)] / 360

: : are those correct?

: : if Mr K says "yes", hope I've helped you Kurt!!
: ************************************
: V_1 = R^3/(12*pi) * T^2

: V_2 = R^3/(12*pi) *(2pi - T)^2

: V = V_1 + V_2 = V_2 = R^3/(12*pi) *(T)^2
: + R^3/(12*pi) *(2pi - T)^2

: dV/dT = R^3/(12*pi) * [2T - 2*(2pi -T)] = 0 ... for maximum

: [2T - 2*(2pi -T)] = 0

: T = pi ......[or 180°]

: Mr. B as the problem is presented, the heights of the cones are equal - the radius of the original circle.
********************************
The heights of the cones are not equal -- their "slant" heights are equal (=R)

radius of the base of small cone = = R * T/(2*pi)

height of the small cone = sqrt[R^2 - {R * T/(2*pi)}^2]

=R/(2pi)*sqrt[(2pi)^2 - T^2]

radius of the base of large cone = = R * (2pi-T)/(2*pi)

height of the large cone = sqrt[R^2 - {R * (2pi-T)/(2*pi)}^2]

Now use all this "mess" to calculate V_1 and V_2. Remember herewe are dealing with radians (not degrees). So pi = 3.14159... (not 180°)

Follow Ups:

Mr B you are partially correct : I think I'm FULLY correct.... Denis Borris 12:53:27 10/09/02 (4)
- Hokay.... Denis Borris 16:26:44 10/09/02 (3)
  - Please check your equation... Subhotosh Khan 16:53:07 10/09/02 (2)
    - NO! Denis Borris 18:14:54 10/09/02 (1)
      - Yes .... Subhotosh Khan 08:54:59 10/10/02 (0)

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: : : : : I understand what needs to be done, Mr K, but can't finish it. : : : : : Too many things I haven't seen before. : : : : : :mAOB = T : : : : : I understand "AOB", but what is T? degrees, like 75 degrees? Yes : : : : : What does "m" mean? It is short of "measure of" or "value of" angle AOB : : : : : :From that you can get the area of the base(A_1) : : : : : Why do I need to get the area? Look below : : : : : :volume of small cone (V_1)=1/3*R*A : : : : : where does that come from? V = (pi r^2 h) / 3, is it not? In this case A = pi * r^2 .... and h = R (the radius of the circle you started with) so we are talking about the same thing : : : : *********************************** : : : : Assume mAOB = T (in radians) : : : : So chord AB = R * T (definition of angle) : : : : So the circumference of the base of the small cone : : : : 2* pi * r = R * T ? (r = radius of the circular base of the small cone) : : : : r = R * T/(2*pi) : : : : V_1 = (pi)* r^2 * h / 3 : : : : = (pi)* [R * T/(2*pi)]^2 * R / 3 : : : : = R^3/(12*pi) * T^2 : : : : Similarly find V_2 : : : was fooling around with this; interim question, Mr K (I'm rusty with cones!): : : : height of smaller cone = Rsqrt(360^2 - T^2) / 360 : : : height of bigger cone = Rsqrt[T(720 - T)] / 360 : : : are those correct? : : : if Mr K says "yes", hope I've helped you Kurt!! : : ************************************ : : V_1 = R^3/(12*pi) * T^2 : : V_2 = R^3/(12*pi) *(2pi - T)^2 : : V = V_1 + V_2 = V_2 = R^3/(12*pi) *(T)^2 : : + R^3/(12*pi) *(2pi - T)^2 : : dV/dT = R^3/(12*pi) * [2T - 2*(2pi -T)] = 0 ... for maximum : : [2T - 2*(2pi -T)] = 0 : : T = pi ......[or 180°] : : Mr. B as the problem is presented, the heights of the cones are equal - the radius of the original circle. : ******************************** : The heights of the cones are not equal -- their "slant" heights are equal (=R) : radius of the base of small cone = = R * T/(2*pi) : height of the small cone = sqrt[R^2 - {R * T/(2*pi)}^2] : =R/(2pi)*sqrt[(2pi)^2 - T^2] : radius of the base of large cone = = R * (2pi-T)/(2*pi) : height of the large cone = sqrt[R^2 - {R * (2pi-T)/(2*pi)}^2] : Now use all this "mess" to calculate V_1 and V_2. Remember herewe are dealing with radians (not degrees). So pi = 3.14159... (not 180°)

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