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Math Help - Circular Cone

  1. #1
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    Exclamation Circular Cone

    Consider a right circular cone with the vertex V and O the center of the base. the radius of the base is 1 and the height VO is square root of 35. consider also the diameter AB of the base and C the midpoint of VA. Comput the length of the shortest path, on the surface of the cone from B to C.
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  2. #2
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    Hello, Ilsa!

    Consider a right circular cone with the vertex V and the center of the base O.
    The radius of the base is 1 and the height VO is \sqrt{35}
    Consider also the diameter AB of the base and C the midpoint of VA.

    Compute the length of the shortest path, on the surface of the cone from B to C.
    Code:
                V
                *
               /|\
              / | \
             /  |  \
          C * __|   \ 6
           / √35|    \
          /     |     \ 
         /      |      \
      A * - - - * - - - * B
                O   1
    We have a right triangle with legs 1 and \sqrt{35}.
    The hypotenuse (slant height of the cone) is 6.


    If we "unroll" half the cone and press it flat,
    . . we have the diagram below.


    Code:
                        V
                        *
                       / \ 
                      /30\ 3
                     /     \  C
                  6 /       * 
                   /     *   \
                  /   *       \ 3
                 / *           \
              B * - - - - - - - *
                      * * *
                        π

    The circumference of the base of the cone is: . 2\pi r \:=\:2\pi(1) \:=\:2\pi
    Half the circumference is: \pi

    We have: . s \:=\:r\theta \quad\Rightarrow\quad \theta \:=\:\frac{s}{r}
    . . Hence: . \theta \:=\:\frac{\pi}{6}
    The vertex angle is 30.


    The shortest distance between B and C is a straight line.


    In \Delta BVC, Law of Cosines:
    . . BC^2 \:=\:6^2 + 3^2 - 2(6)(3)\cos30^o \;=\;45 - 36\left(\frac{\sqrt{3}}{2}\right) \;=\;9(5 - 2\sqrt{3})

    Therefore: . BC \;=\;3\sqrt{5-2\sqrt{3}} \;\approx\;3.718

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