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A cylinder inscribed in a cone with cone H = 8 and cylinder h = 4

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a) Find the ratio of cone-volume / inscribed-volume. Solution:

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b) Find the ratio of cone-area / inscribed-area ?

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closed as off-topic by Goos, Michael Hoppe, Davide Giraudo, M Turgeon, egreg Feb 27 at 14:22

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Please tell us more about what you want help with and what you've tried so far. –  naslundx Feb 27 at 13:01
    
In your solution you've written $H-h = 4$, but should it not be $3$? –  naslundx Feb 27 at 13:24

1 Answer 1

If you know the radius of the base of the cone is $R$ and height $H=8$, by the pythagorean theorem the slant $S = \sqrt{R^2 + 8^2}$.

By the formulae for surface area of cylinder and cone, we have

$$A_{cone} = \pi R(R+S) = \pi R\left(R + \sqrt{R^2 + 64}\right)$$

$$A_{cylinder} = 2 \pi r(r + h) = 2 \pi \frac{R}{2}\left(\frac{R}{2} + 4\right) = \pi R \left(\frac{R}{2} + 4\right)$$

Divide the two expressions and simplify to get the ratio of the two quantities:

$$\frac{A_{cone}}{A_{cylinder}}=\frac{\pi R\left(R + \sqrt{R^2 + 64}\right)}{\pi R \left(\frac{R}{2} + 4\right)}=\frac{R + \sqrt{R^2 + 64}}{\frac{R}{2} + 4}$$

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That's why it's good to tell us more about how far you've managed yourself. I added the final step, does it make sense? –  naslundx Feb 27 at 14:11
    
Since we only know the heights, there is not enough information to calculate $R$. –  naslundx Feb 27 at 14:21
    
The expression is not constant so it cannot be simplified to an expression without the variable $R$. I would say the area ratio is dependant on the radius $R$. –  naslundx Feb 27 at 14:27
    
That's great, thank u :)) –  whyguy Feb 27 at 14:34

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