Practice Questions

A right circular cone has a height of 8 cm and a base radius of 6 cm. Calculate its slant height.

What is the formula for the curved surface area of a hemisphere with radius $r$?

List the formulas for the curved surface area and total surface area of a hemisphere of radius $r$.

State the formulas for the total surface area of a cone and the volume of a cone.

The curved surface area of a right circular cone is $188.4 \text{ cm}^2$ and its slant height is 10 cm. Find the radius of the base. (Use $\pi = 3.14$)

Justify why the total surface area of a hemisphere is $3\pi r^2$ and not half the surface area of a sphere.

Recall the formula for the volume of a right circular cone and identify each variable in the formula.

Summarize the formulas for the surface area and volume of a sphere with radius $r$.

State the formula for the volume of a sphere with radius $r$.

Write the formula for the total surface area of a right circular cone.

Define the slant height of a cone.

List the formulas for: (i) Curved surface area of a cone (ii) Total surface area of a hemisphere (iii) Volume of a sphere (iv) Surface area of a sphere (v) Volume of a cone Explain what each variable ($r, h, l$) represents.

Calculate the surface area of a sphere with a diameter of 14 cm. (Use $\pi = \frac{22}{7}$)

Calculate the total surface area of a solid hemisphere with a radius of 7 cm. (Use $\pi = \frac{22}{7}$)

A student claims that if the radius of a sphere is doubled, its volume becomes four times larger. Critique this statement and provide a mathematical justification.

The radii of two spheres are in the ratio 2:3. Analyze this ratio to find the ratio of their surface areas and the ratio of their volumes.

Identify the relationship between the height ($h$), radius ($r$), and slant height ($l$) of a right circular cone using the Pythagoras theorem.

Explain the difference between the surface area of a sphere and the total surface area of a hemisphere.

Describe how a right circular cone is generated from a right-angled triangle.

Design a closed cylindrical container that can hold exactly 1 litre ($1000 \text{ cm}^3$) of liquid, such that its height is equal to the diameter of its base. Justify your choice of dimensions by calculating the required radius and height.

Derive a formula for the radius ($r$) of a sphere for which the numerical value of its volume is twice the numerical value of its surface area. Justify each step of the derivation.

Evaluate the ratio of the curved surface area of a cone to the surface area of a sphere, if the cone's slant height is equal to the sphere's diameter and the cone's base radius is equal to the sphere's radius. Provide a simplified final ratio.

The volume of a sphere is $36\pi$ cm³. Analyze the given information to find its radius.

Two cones have the same base radius. The ratio of their heights is 4:5. Compare the ratio of their volumes.

A conical party hat has a base diameter of 14 cm and a height of 24 cm. Calculate the area of the paper sheet required to make one such hat. (Use $\pi = \frac{22}{7}$)

A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of the same radius. The total height of the toy is 15.5 cm. Calculate the total surface area of the toy. (Use $\pi = \frac{22}{7}$)

A solid toy is designed in the form of a hemisphere surmounted by a cone. The height of the cone is equal to the diameter of its base. If the radius of the hemisphere is $r$, formulate an expression for the total volume of the toy in terms of $r$ only. Justify your steps.

A right circular cylinder just encloses a sphere of radius $r$. Justify that the surface area of the sphere is equal to the curved surface area of the cylinder. Then, create a new ratio comparing the total surface area of the cylinder to the surface area of the sphere.

The volume of a right circular cone is $9856 \text{ cm}^3$. If the diameter of the base is 28 cm, find the height of the cone.

Explain the components of the total surface area of a right circular cone. Your explanation should describe both the curved surface and the base, and state the formula for each part.

A solid metallic sphere of radius 10.5 cm is melted and recast into a number of smaller cones, each of radius 3.5 cm and height 3 cm. Justify, through calculation, the exact number of cones that can be formed.

Propose a method to determine the radius of a solid sphere if you are only given a hollow cylinder with the same radius and height equal to the sphere's diameter, and enough water to fill the cylinder.

List all the formulas related to the surface area and volume of a hemisphere. Define all the variables used.

Formulate a single algebraic equation that relates the radius ($r$) and height ($h$) of a cone to the radius ($R$) of a sphere, given that their volumes are equal.

A solid metallic sphere of radius 10.5 cm is melted and recast into smaller cones, each of radius 3.5 cm and height 3 cm. Solve for the number of cones that can be formed.

Evaluate which of the two containers is more efficient in terms of material usage for a closed container: a cube with a volume of $64 \text{ cm}^3$ or a sphere with the same volume. Justify your answer by comparing their surface areas.

A well with a 14 m diameter is dug 10 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 7 m to form an embankment. Analyze the situation to find the height of the embankment.

A right circular cone with radius $r$ and height $h$ is carved out from a solid wooden cube of side length $s$. Propose the conditions on $r$, $h$, and $s$ that would maximize the volume of the cone. Justify your proposal.

A company wants to manufacture ice cream cones. They have two proposed designs. Design A: radius 3 cm, height 10 cm. Design B: radius 4 cm, height 6 cm. Evaluate which design holds more ice cream (volume) and which design requires more paper for its curved surface (area). Justify your conclusions with calculations.

Design a tent that is a cylinder of radius 7 m surmounted by a cone of the same radius. The total height of the tent is 24 m and the height of the cylindrical part is 10 m. Formulate a plan to find the total area of canvas required and the total cost if the canvas costs ₹ 80 per m². Justify all steps in your plan.

A hemispherical bowl is completely filled with water. The contents are emptied into a vertical right circular cone. Propose the dimensions of the cone (in terms of the hemisphere's radius $R$) that would be exactly filled with this water, given that the cone's height is equal to its base radius. Justify your proposed dimensions.

A farmer has a heap of wheat in the form of a cone. The circumference of its base is 44 m and its height is 3 m. Calculate the volume of the wheat. The heap is to be covered by a canvas sheet. Find the area of the canvas required and the cost if the canvas costs ₹120 per m². (Use $\pi = \frac{22}{7}$)

Describe the relationship between the volume of a cone and the volume of a cylinder that have the same base radius and height. Use the formulas to support your description.

A hollow sphere with internal and external diameters of 6 cm and 10 cm respectively is melted and recast into a solid cone of base diameter 10 cm. Determine the height and the slant height of the cone.