Surface Areas and Volumes Set-1

Q1. A right circular cylinder has base radius $7\text{ cm}$ and height $24\text{ cm}$. Find its total surface area.

Q2. A solid is formed by joining a hemisphere (flat face) to the base of a right circular cone. The hemisphere has radius $6\text{ cm}$ and the cone has base radius $6\text{ cm}$ and height $8\text{ cm}$. Find the total outer surface area of the resulting solid (exclude the circular joint where they are attached).

Q3. A right circular cone has height equal to twice its base radius. A sphere is inscribed in this cone (touching the base and the lateral surface). Find the ratio of the volume of the sphere to the volume of the cone.

Q4. A right circular cone is cut by a plane parallel to its base creating a smaller top cone and a frustum below. If the curved surface area of the frustum equals the curved surface area of the smaller (top) cone, find the ratio of the height of the smaller cone to the height of the frustum.

Q5. A hemisphere and a right circular cone have the same base radius $r$ and equal volumes. Find the ratio of the total surface area of the hemisphere (including its circular base) to the total surface area of the cone (including its base). (Use $V_{\text{hemisphere}}=\tfrac{2}{3}\pi r^3$, $V_{\text{cone}}=\tfrac{1}{3}\pi r^2h$, and total area of hemisphere = $3\pi r^2$.)

Q6. A solid sphere of radius $6\ \text{cm}$ is melted and recast into a right circular cone with base radius $9\ \text{cm}$. What is the height of the cone?

Q7. A solid is formed by attaching a hemisphere to one circular face of a right circular cylinder whose height equals its diameter. If the total volume of the solid is $72\pi\ \text{cm}^3$, what is the radius (in cm) of the cylinder and hemisphere?

Q8. A spherical shell has inner radius $r$ and outer radius $2r$. What is the ratio of the volume of the material of the shell to the volume of the hollow (empty) inner sphere?

Q9. A right circular cone has a fixed slant height $l=10\ \text{cm}$. For what value of its vertical height $h$ is its volume maximum? (Express your answer in simplest radical form.)

Q10. A right circular cone and a hemisphere have the same base radius $r$. Their curved surfaces are equal in area. If they are treated as separate solids, what is the ratio of the volume of the cone to the volume of the hemisphere?

Q11. A solid hemisphere of radius $8\ \text{cm}$ is melted and recast into identical solid spheres each of radius $4\ \text{cm}$. How many such spheres can be made?

Q12. A right circular cone of height $12\ \text{cm}$ and base radius $5\ \text{cm}$ has its top portion removed by cutting with a plane parallel to the base at a distance $6\ \text{cm}$ from the apex, leaving a frustum. Find the curved surface area of the remaining frustum.

Q13. A solid sphere of radius $12\ \text{cm}$ is melted and recast into hollow right circular cylinders. Each hollow cylinder has inner radius $3\ \text{cm}$, outer radius $5\ \text{cm}$ and height $6\ \text{cm}$. How many such hollow cylinders can be made?

Q14. A plane parallel to the base of a right circular cone of height $H$ cuts the cone into a smaller cone (near the apex) and a frustum. If the curved surface area of the smaller cone equals the curved surface area of the frustum, the distance from the apex to the cutting plane equals

Q15. A solid right circular cylinder of radius $3\ \text{cm}$ and height $10\ \text{cm}$ is closed at both ends. A conical hole of depth $4\ \text{cm}$ and base radius $3\ \text{cm}$ is drilled centrally from one circular face (the entire circular face is removed by the drilling). Find the total surface area of the remaining solid, including the inner conical surface.