Question

Please help. Calculate the volume of the cone and cylinder. Calculate the volume of the shaded region.

Answer 1

notice the picture above, the cone, is inside the cylinder

now, the cylinder has a height of h = 9, and radius of r = 3
so its volume is
`\bf \textit{volume of a cylinder}\\\\ V=\pi r^2 h\implies V=\pi 3^3\cdot 9\implies V=243\pi`

now, what's the height of the cone? well, the cone is snug-fit inside the cylinder, so, it has a height of h = 9, and a radius r = 3, as well


`\bf \textit{volume of a cone}\\\\ V=\cfrac{\pi r^2 h}{3}\implies V=\cfrac{\pi 3^3\cdot 9}{3}\implies V=\cfrac{243\pi }{3}`

now, bear in mind the area of the cylinder, includes everything inside the cylinder, that means the cone along as well

so, if you subtract the area of the cone, from the area of the cylinder, you'd be making a "hole" in the cylinder, and what's leftover, is the shaded area.

Answer 2

The volume of the shaded region is 169.56 m³

The volume of the shaded area is :

• Area of cylinder - Area of cone

Using the parameters given ;

• height = 9 m
• radius = 3 m

The volume of the cylinder is ;

• πr²h

Volume = π × 3² × 9 = 254.34 m³

The Volume of the cone is ;

• 1/3πr²h

Volume of cone = 1/3 × π × 3² × 9 = 84.78 m³

The volume of the sahded region :

• 254.34 - 84.78 = 169.56 m³

Hence, the volume of the shaded region is 169.56 m³