A solid consists of a cone on top of a cylinder with a radius equal to that of the cone. The height of the cone is #9 # and the height of the cylinder is #15 #. If the volume of the solid is #48 pi#, what is the area of the base of the cylinder?

1 Answer
Jan 4, 2018

See a solution process below:

Explanation:

The formula for the volume of a cylinder is:

#V_c = pir^2h_c#

The formula for the volume of a cone is:

#V_n = pir^2h_n/3#

We can now write an equation for the volume of the complete solid as:

#V_c + V_n = 48pi#

#pir^2h_c + pir^2h_n/3 = 48pi#

Where:

  • #r# is the radius of the cone and the cylinder.
  • #h_c# is the height of the cylinder
  • #h_n# is the height of the cone

We can substitute what we know from the problem and solve for #r#:

#(pir^2 xx 15) + (pir^2 xx 9/3) = 48pi#

#15pir^2 + 3pir^2 = 48pi#

#(15 + 3)pir^2 = 48pi#

#18pir^2 = 48pi#

#(18pir^2)/(color(red)(18)color(red)(pi)) = (48pi)/(color(red)(18)color(red)(pi))#

#(color(red)(cancel(color(black)(18pi)))r^2)/cancel(color(red)(18)color(red)(pi)) = (48color(red)(cancel(color(black)(pi))))/(color(red)(18)cancel(color(red)(pi)))#

#(color(red)(cancel(color(black)(18pi)))r^2)/cancel(color(red)(18)color(red)(pi)) = (48color(red)(cancel(color(black)(pi))))/(color(red)(18)cancel(color(red)(pi)))#

#r^2 = 48/color(red)(18)#

#r^2 = (6 xx 8)/color(red)(6 xx 3)#

#r^2 = (color(red)(cancel(color(black)(6))) xx 8)/color(red)(color(black)(cancel(color(red)(6))) xx 3)#

#r^2 = 8/3#

#sqrt(r^2) = sqrt(8/3)#

#r = sqrt(8/3)#

The base of a cylinder is a circle. The formula for the area of a circle is:

#A = pir^2#

We can substitute the radius we calculate above to determine the area of the circle which is the same as the area of the base of the cylinder:

#A = pi(sqrt(8/3))^2#

#A = pi8/3#

#A = 8/3pi#