Question and Answers Forum

All Questions      Topic List

Differentiation Questions

Previous in All Question      Next in All Question      

Previous in Differentiation      Next in Differentiation      

Question Number 97463 by john santu last updated on 08/Jun/20

Commented by bobhans last updated on 08/Jun/20

$$\frac{4}{\mathrm{t}-4}=\frac{\mathrm{R}}{\sqrt{{\mathrm{R}}^2+{\mathrm{t}}^2}}\Rightarrow 16({\mathrm{R}}^2+{\mathrm{t}}^2)={\mathrm{R}}^2({\mathrm{t}}^2-8\mathrm{t}+16)$$$${16\mathrm{t}}^2={\mathrm{R}}^2{\mathrm{t}}^2-{8\mathrm{R}}^2\mathrm{t}\Rightarrow \mathrm{t}=\frac{{8\mathrm{R}}^2}{{\mathrm{R}}^2-16}$$$$\mathrm{volume}\mathrm{v}\left(\mathrm{R}\right)=\frac{1}{3}\pi {\mathrm{R}}^2\left(\frac{{8\mathrm{R}}^2}{{\mathrm{R}}^2-16}\right)$$$$=\frac{8\pi }{3}\left(\frac{{\mathrm{R}}^4}{{\mathrm{R}}^2-16}\right)\Rightarrow \mathrm{v}{}^{\prime }\left(\mathrm{R}\right)=\frac{{4\mathrm{R}}^3({\mathrm{R}}^2-16)-2\mathrm{R}\left({\mathrm{R}}^4\right)}{{({\mathrm{R}}^2-16)}^2}=0$$$$\Rightarrow 2({\mathrm{R}}^2-16)-{\mathrm{R}}^2=0\Rightarrow {\mathrm{R}}^2=32$$$$\mathrm{so}\min \mathrm{vol}=\frac{8\pi }{3}\left(32\right)\left(\frac{32}{16}\right)=\frac{512\pi }{3}$$$$\mathrm{where}\mathrm{t}=\frac{8\times 32}{16}=16.$$

Commented by john santu last updated on 08/Jun/20

$$\mathrm{yes}...\mathrm{correct}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com