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Math Help - Cylinder in a Cone

  1. #1
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    Cylinder in a Cone

    Inscribed a right circular cylinder of height h and radius r in a cone of fixed radius R and fixed height H. Express the volume V of the cylinder as a function of r.

    HINTS GIVEN:

    (1) V = pi(r^2)(h).

    (2) There are similar triangles in the cone.
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  2. #2
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    Quote Originally Posted by symmetry View Post
    Inscribed a right circular cylinder of height h and radius r in a cone of fixed radius R and fixed height H. Express the volume V of the cylinder as a function of r. ...
    Hello,

    I've attached a diagram (see attachment)

    You have 2 similar right triangles:
    - the big one with the legs H and R
    - the small on with the legs (H-h) and r.

    Set up the proportion:

    \frac{r}{R}=\frac{H-h}{H}. Solve for r:

    r=\frac{H-h}{H} \cdot R=R-\frac{R}{H} \cdot h

    Plug in this term into the formula of the volume of a cylinder: V=\pi \cdot r^2 \cdot h

    V(h)=\pi \cdot \left(R-\frac{R}{H} \cdot h  \right)^2 \cdot h

    EB
    Attached Thumbnails Attached Thumbnails Cylinder in a Cone-zylind_im_kegl.gif  
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  3. #3
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    ok

    Absolutely amazing reply and great math notes for my state exam.

    Thanks!
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