The ratio of the altitude of the cone of grea | vedclass

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

Question

(A) Let $h$ be the height of the cone inscribed in a sphere of radius $R$. Let the center of the sphere be $O$. The distance from the center $O$ to th

Vedclass · Accepted Answer

$2/3$

Vedclass · Answer

(A) Let $h$ be the height of the cone inscribed in a sphere of radius $R$. Let the center of the sphere be $O$. The distance from the center $O$ to the base of the cone is $|h-R|$.By the Pythagorean theorem,the radius $r$ of the base of the cone satisfies $r^2 + (h-R)^2 = R^2$,so $r^2 = R^2 - (h^2 - 2hR + R^2) = 2hR - h^2$.The volume $V$ of the cone is $V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (2hR - h^2) h = \frac{1}{3} \pi (2h^2R - h^3)$.To maximize $V$,we find the derivative with respect to $h$: $\frac{dV}{dh} = \frac{1}{3} \pi (4hR - 3h^2)$.Setting $\frac{dV}{dh} = 0$ gives $h(4R - 3h) = 0$. Since $h 
eq 0$,we have $h = \frac{4R}{3}$.The diameter of the sphere is $D = 2R$.The ratio of the altitude of the cone to the diameter of the sphere is $\frac{h}{D} = \frac{4R/3}{2R} = \frac{4}{6} = \frac{2}{3}$.

The ratio of the altitude of the cone of greatest volume which can be inscribed in a given sphere to the diameter of the sphere is:

Similar Questions

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius $R$ is $\frac{2 R}{\sqrt{3}} .$ Also find the maximum volume.

Two stones are thrown vertically upwards with equations of motion $s_1 = 19.6t - 4.9t^2$ and $s_2 = 9.8t - 4.9t^2$ respectively. The maximum height of the first stone is $h$. When the first stone is at its maximum height,the height of the second stone is:

If $x + y = 8$,find the maximum value of $xy$.

If $PQ$ and $PR$ are the two sides of a triangle,then the angle between them which gives the maximum area of the triangle is

If the functions $f(x) = \frac{x^3}{3} + 2bx + \frac{ax^2}{2}$ and $g(x) = \frac{x^3}{3} + ax + bx^2$,where $a \neq 2b$,have a common extreme point,then $a + 2b + 7$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module