The numerical values of the volume and the ar | vedclass

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

Question

(B) The volume of a right circular cone is given by $V = \frac{1}{3} \pi r^{2} h$.The lateral surface area $(LSA)$ of a right circular cone is given b

Vedclass · Accepted Answer

$\frac{1}{9}$

Vedclass · Answer

(B) The volume of a right circular cone is given by $V = \frac{1}{3} \pi r^{2} h$.The lateral surface area $(LSA)$ of a right circular cone is given by $LSA = \pi r l$,where $l = \sqrt{r^{2} + h^{2}}$ is the slant height.According to the problem,the numerical values of the volume and the lateral surface area are equal:$\frac{1}{3} \pi r^{2} h = \pi r \sqrt{r^{2} + h^{2}}$Dividing both sides by $\pi r$ (assuming $r 
eq 0$):$\frac{1}{3} r h = \sqrt{r^{2} + h^{2}}$Squaring both sides:$\frac{r^{2} h^{2}}{9} = r^{2} + h^{2}$Dividing both sides by $r^{2} h^{2}$:$\frac{1}{9} = \frac{r^{2} + h^{2}}{r^{2} h^{2}}$$\frac{1}{9} = \frac{r^{2}}{r^{2} h^{2}} + \frac{h^{2}}{r^{2} h^{2}}$$\frac{1}{9} = \frac{1}{h^{2}} + \frac{1}{r^{2}}$Therefore,the value of $\frac{1}{h^{2}} + \frac{1}{r^{2}}$ is $\frac{1}{9}$.

The numerical values of the volume and the area of the lateral surface of a right circular cone are equal. If the height of the cone is $h$ and the radius is $r,$ the value of $\frac{1}{h^{2}}+\frac{1}{r^{2}}$ is

Similar Questions

$A$ field is in the form of a rectangle of length $18\, m$ and width $15\, m$. $A$ pit,$7.5\, m$ long,$6\, m$ broad and $0.8\, m$ deep,is dug in a corner of the field and the earth taken out is evenly spread over the remaining area of the field. The level of the field raised is......$cm$

The sum of the radius of the base and the height of a solid cylinder is $37\, m$. If the total surface area of the solid cylinder is $1628\, m^2$,then the circumference of its base and the volume of the cylinder are:

The diagonal of a cubical box is $\sqrt{300} \text{ cm}$. Find out the surface area.

Water is flowing at the rate of $5 \text{ km/h}$ through a pipe of diameter $14 \text{ cm}$ into a rectangular tank which is $50 \text{ m}$ long and $44 \text{ m}$ wide. The time taken,in hours,for the rise in the level of water in the tank to be $7 \text{ cm}$ is:

The height of a cone is $30 \, cm$. $A$ small cone is cut off at the top by a plane parallel to the base. If its volume is $\frac{1}{27}$ of the volume of the given cone,at what height above the base is the section made? (in $cm$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module