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Question Number 175568 by ajfour last updated on 02/Sep/22

Commented by ajfour last updated on 02/Sep/22

$$Findthemaximuminnercone$$$$volumeifouterconevolumeis$$$$unity.$$

Answered by mr W last updated on 02/Sep/22

$$outercone:R,H,V$$$$innercone:r,h,v$$$$V=\frac{\pi R^{2}H}{3}(=1)$$$$v=\frac{\pi r^{2}h}{3}$$$$\lambda =\frac{v}{V}=\left(\frac{h}{H}\right){\left(\frac{r}{R}\right)}^2$$$$\frac{r}{R}=\frac{H-h}{H}=1-\frac{h}{H}$$$$with\xi =\frac{h}{H}<1$$$$\Rightarrow \lambda =\xi {(1-\xi )}^2$$$$\frac{d\lambda }{d\xi }={(1-\xi )}^2-2\xi (1-\xi )=0\Rightarrow \xi =\frac{1}{3}$$$${\lambda }_{max}=\frac{1}{3}{(1-\frac{1}{3})}^2=\frac{4}{27}\approx 0.148✓$$

Commented by ajfour last updated on 03/Sep/22

$$goodquestion,perfectsolution.$$$$thankyousir!$$

Commented by mr W last updated on 03/Sep/22

$$indeedaverygoodquestion!$$

Commented by Tawa11 last updated on 15/Sep/22

$$\mathrm{Great}\mathrm{sir}$$

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