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 #7 #. If the volume of the solid is #42 pi#, what is the area of the base of the cylinder?

Question

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 #7   #.  If the volume of the solid is #42 pi#, what is the area of the base of the cylinder?

Maverick Smith · Accepted Answer

The area of the base #=4.2pi=13.2 (unit)^2#

Volume of the cone is #=1/3*a*h#

Where,

#a=#area of the base

#h=# height of the cone #=9#

Volume of the cylinder is #=a*H#

where

#H=# height of the cylinder #=7#

Total amount

#V=1/3*a*h+a*H=42pi#

Consequently,

#a(h/3+H)=42pi#

#a(9/3+7)=42pi#

#10a=42pi#

#a=42/10pi=4.2pi=13.2 (unit)^2#

Mason Adams · Answer

undefined To find the area of the base of the cylinder, we first need to find the volume of both the cone and the cylinder, and then subtract the volume of the cone from the total volume of the solid to find the volume of the cylinder. Then, using the formula for the volume of a cylinder, we can find the radius of the cylinder. Finally, we use the formula for the area of the base of a cylinder to find the area of the base.

Given:
Height of the cone (h₁) = 9 units
Height of the cylinder (h₂) = 7 units
Total volume of the solid = 42π cubic units

Volume of cone (V₁) = (1/3)πr₁²h₁
Volume of cylinder (V₂) = πr₂²h₂

Total volume of solid = V₁ + V₂
42π = (1/3)πr₁²h₁ + πr₂²h₂

Since the radius of the cone (r₁) is equal to the radius of the cylinder (r₂), we can substitute r for both.

42π = (1/3)πr²(9) + πr²(7)
42π = (3πr² + 7πr²)
42π = 10πr²

Divide both sides by 10π:
4.2 = r²

Take the square root of both sides to find the radius of the cylinder:
r = √4.2

Now that we have the radius of the cylinder, we can find the area of its base using the formula:
Area of base = πr²

Substitute the value of r into the formula:
Area of base = π(√4.2)²
Area of base ≈ π(4.2)
Area of base ≈ 4.2π square units

Therefore, the area of the base of the cylinder is approximately 4.2π square units.