The volume of the largest possible right circ | vedclass

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

Question

(C) Let the radius of the sphere be $R = \sqrt{3}$. Let the height of the cylinder be $h$ and its radius be $r$.In the right-angled triangle formed by

Vedclass · Accepted Answer

$4\pi$

Vedclass · Answer

(C) Let the radius of the sphere be $R = \sqrt{3}$. Let the height of the cylinder be $h$ and its radius be $r$.In the right-angled triangle formed by the radius of the sphere,the radius of the cylinder,and half the height of the cylinder,we have:$R^2 = r^2 + (h/2)^2$$(\sqrt{3})^2 = r^2 + \frac{h^2}{4}$$3 = r^2 + \frac{h^2}{4} \Rightarrow r^2 = 3 - \frac{h^2}{4}$The volume of the cylinder is $V = \pi r^2 h = \pi (3 - \frac{h^2}{4})h = 3\pi h - \frac{\pi h^3}{4}$.To maximize the volume,we find the derivative with respect to $h$ and set it to zero:$\frac{dV}{dh} = 3\pi - \frac{3\pi h^2}{4} = 0$$3\pi = \frac{3\pi h^2}{4} \Rightarrow h^2 = 4 \Rightarrow h = 2$.Substituting $h = 2$ into the volume formula:$V = \pi (3 - \frac{2^2}{4})(2) = \pi (3 - 1)(2) = 4\pi$.

The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius $R = \sqrt{3}$ is

Similar Questions

If an open cylinder of given surface area has maximum volume,then its radius is

The function $f(x) = x^5 - 5x^4 + 5x^3 - 10$ has a local maximum at $x =$

The maximum area of a rectangle inscribed in a circle of radius $10 \text{ cm}$ is

The maximum value of the function $f(x) = -|x+1| + 3, x \in R$ is . . . . . . .

If $f(x)=x^2+ax+b$ has a minima at $x=3$ whose value is $5$,then the values of $a$ and $b$ are respectively.

Vedclass Test Series

Exam Paper Generator

Online Exam Module