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 27 27 and the height of the cylinder is 7 7. If the volume of the solid is 96 pi96π, what is the area of the base of the cylinder?
1 Answer
Explanation:
The volume of the cylinder is given by its height multiplied by the area of its circular base.
V_"cylinder" = pi * r^2 * h_"cylinder"Vcylinder=π⋅r2⋅hcylinder
h_"cylinder" = 7hcylinder=7 in this question.
The volume of a cone is given by a third of its height multiplied by the area of its circular base.
V_"cone" = 1/3 * pi * r^2 * h_"cone"Vcone=13⋅π⋅r2⋅hcone
h_"cone" = 27hcone=27 in this question.- The variable
rr is reused as the cone has the same radius as the cylinder.
The volume of the entire solid is
V_"solid" = V_"cylinder" + V_"cone"Vsolid=Vcylinder+Vcone
= pi * r^2 * h_"cylinder" + 1/3 * pi * r^2 * h_"cone"=π⋅r2⋅hcylinder+13⋅π⋅r2⋅hcone
= pi * r^2 * (h_"cylinder" + 1/3 h_"cone")=π⋅r2⋅(hcylinder+13hcone)
= pi * r^2 * (7 + 1/3 xx 27)=π⋅r2⋅(7+13×27)
Now it becomes a simple matter to solve for the base area of the cylinder, which is just
pi * r^2 = V_"solid"/(7 + 1/3 xx 27)π⋅r2=Vsolid7+13×27
= (96pi)/16=96π16
= 6pi=6π
~~ 18.850≈18.850