A sector is removed from a metallic disc and | vedclass

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

Question

(B) Let the radius of the metallic disc be $R$. When a sector is removed and the remaining part is bent to form a cone,the radius $R$ of the disc beco

Vedclass · Accepted Answer

$6$

Vedclass · Answer

(B) Let the radius of the metallic disc be $R$. When a sector is removed and the remaining part is bent to form a cone,the radius $R$ of the disc becomes the slant height $l$ of the cone.Let the radius of the base of the cone be $x$ and its height be $h$.We have the relation $R^2 = x^2 + h^2$,where $R = l$.The volume of the cone is given by $V = \frac{1}{3} \pi x^2 h = 2 \sqrt{3} \pi$.Thus,$x^2 h = 6 \sqrt{3}$,which implies $x^2 = \frac{6 \sqrt{3}}{h}$.Substituting this into the expression for $R^2$:$R^2 = \frac{6 \sqrt{3}}{h} + h^2$.To find the minimum diameter,we minimize $R^2$ with respect to $h$:$\frac{d(R^2)}{dh} = -\frac{6 \sqrt{3}}{h^2} + 2h$.Setting $\frac{d(R^2)}{dh} = 0$,we get $2h = \frac{6 \sqrt{3}}{h^2}$,so $h^3 = 3 \sqrt{3}$.This gives $h = \sqrt{3}$.Then $x^2 = \frac{6 \sqrt{3}}{\sqrt{3}} = 6$.Therefore,$R^2 = 6 + (\sqrt{3})^2 = 6 + 3 = 9$,so $R = 3$.The diameter of the disc is $2R = 2 	imes 3 = 6$.

$A$ sector is removed from a metallic disc and the remaining region is bent into the shape of a circular conical funnel with volume $2 \sqrt{3} \pi$. The least possible diameter of the disc is

Similar Questions

Find the local maximum and local minimum values of the function given by $f(x) = x^{3} - 6x^{2} + 9x + 15$.

The lateral edge of a regular rectangular pyramid is $a \text{ cm}$ long. The lateral edge makes an angle $\alpha$ with the plane of the base. The value of $\alpha$ for which the volume of the pyramid is greatest,is

The function $f(x)=2|x|+|x+2|-||x+2|-2|x||$ has a local minimum or a local maximum at $x=$

If $x+y=60$,$x>0$,$y>0$,then the maximum value of $x y^3$ is

At what value of $x$ does the function $f(x) = x^x$ $(x > 0)$ attain its minimum value?

Vedclass Test Series

Exam Paper Generator

Online Exam Module