If the volume of a right circular cone is reduced by 15% by reducing its height by 5%

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Sep 13th 2013, 05:49 PM
alden35

If the volume of a right circular cone is reduced by 15% by reducing its height by 5%

If the volume of a right circular cone is reduced by 15% by reducing its height by 5%, by what percent must the radius of the base reduced?
Sep 13th 2013, 07:23 PM
chiro

Re: If the volume of a right circular cone is reduced by 15% by reducing its height b

Hey alden35.

Hint: Express the volumes and height of the two cones in terms of r, h, dr, and dh where dr is the change of radius and dh is the change of height.
Sep 14th 2013, 02:48 PM
Soroban

Re: If the volume of a right circular cone is reduced by 15% by reducing its height b

Hello, alden35!

Quote:

If the volume of a right circular cone is reduced by 15% by reducing its height by 5%,
by what percent must the radius of the base reduced?

$\text{We have: }\:V \:=\:\tfrac{\pi}{3}r^2h\;\;[1]$

$\text{Then: }\:0.85V \;=\;\tfrac{\pi}{3}(xr)^2(0.95h)\;\;[2]$

$\text{Divide }[2]\text{ by }[1]\!:\;\frac{0.85V}{V}\;=\;\frac{\frac{\pi}{3}(xr)^2 (0.95h)}{\frac{\pi}{3}r^2h} \quad\Rightarrow\quad 0.85 \:=\:x^2(0.95)$

. . $x^2 \:=\:\frac{0.85}{0.95} \quad\Rightarrow\quad x \:=\:\sqrt{\frac{17}{19}} \:=\: 0.945905303$

$\text{Hence: }\:x \:\approx\:0.946 \:=\:94.6\%$

$\text{Therefore, the radius must be reduced by }5.4\%.$
Sep 15th 2013, 12:56 PM
HallsofIvy

Re: If the volume of a right circular cone is reduced by 15% by reducing its height b

Quote:

Originally Posted by chiro

Hey alden35.

Hint: Express the volumes and height of the two cones in terms of r, h, dr, and dh where dr is the change of radius and dh is the change of height.

Or wait for Soroban to do the problem for you!
Sep 15th 2013, 07:55 PM
chiro

Re: If the volume of a right circular cone is reduced by 15% by reducing its height b

Yeah that could also work as well :)