Question 1138274
The altitude of a regular cone is twice its base radius. if its volume is {{{3400}}} cubic meter what is the total surface area? 


Since the base of a cone is a circle, the formula for finding the volume of a cone is:

{{{V=(1/3)pi*r^2*h}}}

given: 
the altitude of a regular cone is twice its base radius:{{{h=2r}}}

{{{volume is V=3400}}}

{{{3400=(1/3)pi*r^2*2r}}}

{{{3400=(2pi/3)r^3 }}}

{{{r^3=3400/(2/3)pi}}}

{{{r^3=(3*3400)/2pi}}}

{{{r^3=5100/pi}}}

{{{r=root(3,5100/pi)}}}

{{{r=11.75}}}


The total surface area of a cone is the sum of the area of its base and the lateral (side) surface.

the formula for the lateral surface area of a right cone is 
{{{L_S_A= pi*r*l}}} , where l is the slant height of the cone 

use {{{r}}} and {{{h}}} to find the slant height {{{l}}}
{{{l=sqrt(r^2+h^2)}}}

{{{l=sqrt(11.75^2+(2*11.75)^2)}}}

{{{l=26.27}}}

{{{L_S_A= pi*11.75*26.27}}}

{{{L_S_A= 969.72cm^2}}}

area of the base: 

{{{B_A=r^2pi}}}

{{{B_A=11.75^2*pi}}}

{{{B_A=433.74cm^2}}}


=>The total surface area of a cone is:

{{{A=B_A+L_S_A}}}

{{{A=433.74cm^2+969.72cm^2}}}

{{{A=1403.46cm^2}}}