Question

In a regular triangular prism the distance from the centre of one base to one of the vertices of the other base is $l$. The altitude of the prism for which the volume is greatest :

• A. $\frac{l}{2}$
• B. $\frac{l}{\sqrt{3}}$
• C. $\frac{l}{3}$
• D. $\frac{l}{4}$

Answer 1

Answer: (B)

$AG =\frac{2}{3} \cdot \frac{ a \sqrt{3}}{2}=\frac{ a }{\sqrt{3}}$. Now $l^2=\frac{ a ^2}{3}+ h ^2 ; \operatorname{arca}( A )=\frac{\sqrt{3}}{4} a ^2$
$V ( h )=\frac{\sqrt{3}}{4} a ^2 h =\frac{\sqrt{3}}{4} h \left(l^2- h ^2\right) 3 ; V ^{\prime}( h )=0 \Rightarrow h =\frac{l}{\sqrt{3}} ; V _{\max }=\frac{l^3}{2}$