Question

the curved surface are of a cone is 20π cm2 and it's slane height and radius are consecutive natural number . then volume equals​

Answer 1

Answer:

volume of the cone

$=16\,\pi\,cm^3$

Explanation:

$\text{As per the given data,}$

$\text{Radius of the cone, r= n}$

$\text{Slant height, l=n+1}$

$\boxed{\textbf{Curved surface area of cone=}\pi\,r\,l\,square\,units}$

$\text{We have}$

$\pi\,r\,l=20\,\pi$

$\implies\,n(n+1)=20$

$\implies\,n^2+n-20=0$

$\implies\,(n+5)(n-4)=0$

$\implies\,n=4$ (radius cannot be negative)

$\text{r=4\,cm\:and\:l=5\,cm}$

$\text{We know that,}$

$l^2=r^2+h^2$

$5^2=4^2+h^2$

$25-16=h^2$

$9=h^2$

$\implies\,h=3\,cm$

Now,

$\textbf{Volume of the cone }=\frac{1}{3}\,\pi\,r^2\,h$

$\text{Volume of the cone }=\frac{1}{3}\,\pi(4)^2\,(3)$

$\text{Volume of the cone }=16\,\pi\,cm^3$

Answer 2

Answer:

volume of the cone

$=16\,\pi\,cm^3$

Explanation:

$\text{As per the given data,}$

$\text{Radius of the cone, r= n}$

$\text{Slant height, l=n+1}$

$\boxed{\textbf{Curved surface area of cone=}\pi\,r\,l\,square\,units}$

$\text{We have}$

$\pi\,r\,l=20\,\pi$

$\implies\,n(n+1)=20$

$\implies\,n^2+n-20=0$

$\implies\,(n+5)(n-4)=0$

$\implies\,n=4$ (radius cannot be negative)

$\text{r=4\,cm\:and\:l=5\,cm}$

$\text{We know that,}$

$l^2=r^2+h^2$

$5^2=4^2+h^2$

$25-16=h^2$

$9=h^2$

$\implies\,h=3\,cm$

Now,

$\textbf{Volume of the cone }=\frac{1}{3}\,\pi\,r^2\,h$

$\text{Volume of the cone }=\frac{1}{3}\,\pi(4)^2\,(3)$

$\text{Volume of the cone }=16\,\pi\,cm^3$