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Question Number 191454 by Mingma last updated on 24/Apr/23

Commented by mr W last updated on 24/Apr/23

$$xyz=\frac{646}{25}$$

Commented by Mingma last updated on 24/Apr/23

Good!

Answered by mr W last updated on 24/Apr/23

$$\sqrt{4^2-x^2}+\sqrt{5^2-y^2}+\sqrt{6^2-z^2}$$$$⩾\sqrt{{(4+5+6)}^2-{(x+y+z)}^2}$$$$=\sqrt{{15}^2-9^2}$$$$=12$$$$equalityholdsonlyif$$$$\frac{x}{4}=\frac{y}{5}=\frac{z}{6},i.e.x=4k,y=5k,z=6k$$$$(4+5+6)k=9,k=\frac{9}{15}=\frac{3}{5}$$$$\Rightarrow xyz=4\times 5\times 6\times k^3=120\times {\left(\frac{3}{5}\right)}^3=\frac{646}{25}✓$$

Commented by Mingma last updated on 24/Apr/23

Very nice, sir!

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