A-cone-is-placed-inside-a-sphere-If-volume-of-the-cone-is-maximum-find-the-ratio-of-radius-from-the-cone-and-sphere-

Question Number 21840 by Joel577 last updated on 05/Oct/17

A cone is placed inside a sphere .

If volume of the cone is maximum,

find the ratio of radius from the cone and sphere

Answered by mrW1 last updated on 05/Oct/17

R = radius of sphere

r = radius of cone

V = volume of cone

V = \frac{1}{3} π r^{2} \times h = \frac{1}{3} \times π r^{2} \times (R + \sqrt{R^{2} - r^{2}})

V = \frac{π}{3} r^{2} (R + \sqrt{R^{2} - r^{2}})

\frac{dV}{dr} = \frac{2 π r}{3} (R + \sqrt{R^{2} - r^{2}} - \frac{r^{2}}{2 \sqrt{R^{2} - r^{2}}}) = 0

R + \sqrt{R^{2} - r^{2}} - \frac{r^{2}}{2 \sqrt{R^{2} - r^{2}}} = 0

1 + \sqrt{1 - {(\frac{r}{R})}^{2}} - \frac{{(\frac{r}{R})}^{2}}{2 \sqrt{1 - {(\frac{r}{R})}^{2}}} = 0

with x = \frac{r}{R}

1 + \sqrt{1 - x^{2}} - \frac{x^{2}}{2 \sqrt{1 - x^{2}}} = 0

\Rightarrow x \approx 0.943

Commented by Joel577 last updated on 05/Oct/17

I dont understand why V = 4 π r^{2} \sqrt{R^{2} - r^{2}}

I think it is V = \frac{1}{3} π r^{2} h

Commented by mrW1 last updated on 05/Oct/17

you are right sir .

I have corrected .

Commented by Joel577 last updated on 05/Oct/17

t h a n k y o u v e r y m u c h

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