- #1
rxh140630
- 60
- 11
- Homework Statement
- Find the dimensions of the right circular cylinder of maximum volume that can be inscribed in a right circular cone of radius R and altitude H
- Relevant Equations
- [itex] V_{cone}=\frac{\pi}3R^2H[/itex]
[itex]V_{cylinder}={\pi}r^2h[/itex]
Please I do not want the answer, I just want understanding as to why my logic is faulty.
Included as an attachment is how I picture the problem.
My logic:
Take the volume of the cone, subtract it by the volume of the cylinder. Take the derivative. from here I can find the point that the cone will have minimum volume, which will give me the point where the cylinder is at it's maximum volume.
I do not understand why this logic is faulty. Anyways, using the variable in my attachment:
[itex]V_{min}=\frac{\pi}{3}R^2H-{\pi}r^2h=\frac{\pi}{3}R^2H-{\pi}r^{2}(H-y)=\frac{\pi}{3}R^{2}H-{\pi}Hr^2+{\pi}yr^2[/itex]
take derivative of V_min with respect to H
[itex] \frac{\pi}{3}R^2-{\pi}r^2=0 => r=\sqrt{\frac13}R[/itex]
but this is wrong.
Is my logic wrong, or is my math wrong? And if my logic is wrong, why? I don't see why minimizing the volume of the cone is not = maximizing the volume of the cylinder.
Please do not give me the answer.
Included as an attachment is how I picture the problem.
My logic:
Take the volume of the cone, subtract it by the volume of the cylinder. Take the derivative. from here I can find the point that the cone will have minimum volume, which will give me the point where the cylinder is at it's maximum volume.
I do not understand why this logic is faulty. Anyways, using the variable in my attachment:
[itex]V_{min}=\frac{\pi}{3}R^2H-{\pi}r^2h=\frac{\pi}{3}R^2H-{\pi}r^{2}(H-y)=\frac{\pi}{3}R^{2}H-{\pi}Hr^2+{\pi}yr^2[/itex]
take derivative of V_min with respect to H
[itex] \frac{\pi}{3}R^2-{\pi}r^2=0 => r=\sqrt{\frac13}R[/itex]
but this is wrong.
Is my logic wrong, or is my math wrong? And if my logic is wrong, why? I don't see why minimizing the volume of the cone is not = maximizing the volume of the cylinder.
Please do not give me the answer.
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