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Question Number 164941 by ajfour last updated on 23/Jan/22

Commented by ajfour last updated on 23/Jan/22

$$Ifradiiofinscribedandcircum-$$$$scribedspheresare\mathit{a}and\mathit{b},find$$$$heightandradiusofcone.$$

Answered by mr W last updated on 23/Jan/22

$$height=h,radius=r$$$$(2b-h)h=r^2...\left(i\right)$$$$\frac{h-a}{a}=\frac{\sqrt{h^2+r^2}}{r}...\left(ii\right)$$$$from\left(ii\right):$$$$r^2=\frac{{ha}^2}{(h-2a)}$$$$\frac{{ha}^2}{(h-2a)}=(2b-h)h$$$$h^2-2(a+b)h+a(a+4b)=0$$$$\Rightarrow h=a+b+\sqrt{b(b-2a)}$$$$r=a\sqrt{\frac{h}{h-2a}}$$$$\Rightarrow r=a\sqrt{\frac{b+a+\sqrt{b(b-2a)}}{b-a+\sqrt{b(b-2a)}}}$$

Commented by Tawa11 last updated on 23/Jan/22

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

Commented by ajfour last updated on 24/Jan/22

$$Verystraightnpragmatic,$$$$reallygreatsir.Thankyou.$$

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