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Question Number 53450 by ajfour last updated on 22/Jan/19

Commented by ajfour last updated on 22/Jan/19

$$Findradiusofinscribedspherein$$$$box.$$

Answered by ajfour last updated on 22/Jan/19

$$letDbeorigin.xaxisbeDA,$$$$yaxisbeDC,zaxisbeDF.$$$$E(a,0,2p);B(a,2a,0);G(0,2a,p);$$$$F(0,0,3p).$$$$BG\times BE=(-ai+pk)\times (-2aj+2pk)$$$$\Rightarrow \overline{n}=2a^{2}k+2apj+2api$$$$eq.ofceilingplaneis$$$$(\overline{r}-3pk).\overline{n}=0$$$$\Rightarrow (\overline{r}-3p\hat{k}).(p\hat{i}+p\hat{j}+a\hat{k})=0$$$$\Rightarrow \overline{r}.(p\hat{i}+p\hat{j}+a\hat{k})=3ap$$$$centerofsphere:G(r,r,r)$$$$\Rightarrow [r(i+j+k)+\frac{r\overline{n}}{\mid \overline{n}\mid }].(pi+pj+ak)=3ap$$$$\Rightarrow r=\frac{3ap}{2p+a+\sqrt{2p^2+a^2}}.$$

Commented by mr W last updated on 22/Jan/19

$$bestwayinthiscase!$$

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