A cone of maximum volume is inscribed in a sp | vedclass

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

Question

(A) Let the radius of the sphere be $R$. Let the height of the cone be $h$ and the radius of the base of the cone be $x$. From the geometry of the sph

Vedclass · Accepted Answer

$2/3$

Vedclass · Answer

(A) Let the radius of the sphere be $R$. Let the height of the cone be $h$ and the radius of the base of the cone be $x$. From the geometry of the sphere,the distance from the center of the sphere to the base of the cone is $|h - R|$. Using the Pythagorean theorem in the triangle formed by the sphere's radius,the cone's base radius,and the distance from the center,we have $x^2 + (h - R)^2 = R^2$. $x^2 = R^2 - (h^2 - 2hR + R^2) = 2hR - h^2$. The volume of the cone is $V = \frac{1}{3}\pi x^2 h = \frac{1}{3}\pi (2hR - h^2)h = \frac{1}{3}\pi (2Rh^2 - h^3)$. To maximize volume,differentiate with respect to $h$: $\frac{dV}{dh} = \frac{1}{3}\pi (4Rh - 3h^2)$. Setting $\frac{dV}{dh} = 0$,we get $h(4R - 3h) = 0$. Since $h 
eq 0$,$h = \frac{4}{3}R$. The diameter of the sphere is $D = 2R$. The ratio of the height of the cone to the diameter of the sphere is $\frac{h}{D} = \frac{\frac{4}{3}R}{2R} = \frac{4}{6} = \frac{2}{3}$.

$A$ cone of maximum volume is inscribed in a sphere of radius $R$. The ratio of the height of the cone to the diameter of the sphere is:

Similar Questions

On the interval $[0, 1],$ the function $f(x) = {x^{25}}{(1 - x)^{75}}$ takes its maximum value at the point

Let $A$ be the region enclosed by the parabola $y^2=2x$ and the line $x=24$. Then the maximum area of the rectangle inscribed in the region $A$ is ..................

If $f(x) = 2x^3 - 21x^2 + 36x - 30$,then which one of the following is correct?

If $f(x) = 2x^3 - 3x^2 - 12x + 5$ and $x \in [-2, 4]$,then the maximum value of the function occurs at which value of $x$?

If $y = a \log |x| + b x^2 + x$ has its extremum values at $x = -1$ and $x = 2$,then

Vedclass Test Series

Exam Paper Generator

Online Exam Module