In a regular triangular prism,the distance fr | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $a$ be the side length of the equilateral triangular base and $h$ be the altitude of the prism.The distance from the centre $G$ of the base to

Vedclass · Accepted Answer

$\frac{l}{\sqrt{3}}$

Vedclass · Answer

(B) Let $a$ be the side length of the equilateral triangular base and $h$ be the altitude of the prism.The distance from the centre $G$ of the base to a vertex $A$ is the circumradius of the equilateral triangle,given by $R = \frac{a}{\sqrt{3}}$.In the right-angled triangle formed by the altitude $h$,the circumradius $R$,and the distance $l$,we have $l^2 = R^2 + h^2 = \frac{a^2}{3} + h^2$.Thus,$a^2 = 3(l^2 - h^2)$.The volume of the prism is $V = 	ext{Area of base} 	imes h = \left( \frac{\sqrt{3}}{4} a^2 ight) h$.Substituting $a^2$,we get $V(h) = \frac{\sqrt{3}}{4} 	imes 3(l^2 - h^2) h = \frac{3\sqrt{3}}{4} (l^2 h - h^3)$.To maximize $V$,we find $V'(h) = \frac{3\sqrt{3}}{4} (l^2 - 3h^2) = 0$.Setting $V'(h) = 0$ gives $l^2 = 3h^2$,so $h = \frac{l}{\sqrt{3}}$.Thus,the altitude for which the volume is greatest is $\frac{l}{\sqrt{3}}$.

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

Similar Questions

The function $f(x) = ax + \frac{b}{x}$ where $a, b, x > 0$ takes on the least value at $x$ equal to:

$A$ circular sector of perimeter $60 \ m$ with maximum area is to be constructed. The radius of the circular arc in meters must be: (in $m$)

Let $f(x)=3^{(x^{2}-2)^{3}+4}, x \in R$. Then which of the following statements are true?
$P: x=0$ is a point of local minima of $f$
$Q: x=\sqrt{2}$ is a point of inflection of $f$
$R: f^{\prime}$ is increasing for $x>\sqrt{2}$

Given $P(x) = x^4 + ax^3 + bx^2 + cx + d$ such that $x=0$ is the only real root of $P'(x) = 0$. If $P(-1) < P(1)$,then in the interval $[-1, 1]$:

If ${a^2}{x^4} + {b^2}{y^4} = {c^6}$,then the maximum value of $xy$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module