Practice Questions

Explain why, when finding the surface area of a new solid formed by joining two identical cubes end-to-end, we do not simply add their individual total surface areas.

To double the volume of a solid sphere, a student proposes doubling its radius. Justify whether this proposal is correct or incorrect.

Define the Total Surface Area (TSA) of a solid object.

List the two basic solids that are combined to form a medicine capsule as shown in the chapter.

A student argues that to find the surface area of a toy made of a cone mounted on a hemisphere, one must add the Total Surface Area (TSA) of the cone and the TSA of the hemisphere. Critique this argument.

List the formulas required to calculate the total surface area of canvas used for a tent which is in the shape of a cylinder surmounted by a conical top. (Assume the base is not covered with canvas).

A company manufactures two types of containers. Container A is a cylinder with a radius of 7 cm and a height of 14 cm. Container B is a cube with an edge of 12 cm. Justify which container is a more "efficient" use of material by comparing their volume-to-surface-area ratios. (Use $\pi = \frac{22}{7}$)

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

From a solid cylinder of height 10 cm and base radius 4 cm, a conical cavity of the same height and same base radius is hollowed out. Determine the volume of the remaining solid.

State the formula for the Curved Surface Area (CSA) of a cylinder with radius '$r$' and height '$h$'.

Calculate the volume of a solid formed by a cone of radius 3 cm and height 4 cm, mounted on a hemisphere of radius 3 cm.

A wooden article is created by scooping out a hemisphere from each end of a solid cylinder. The height of the cylinder is 15 cm and the radius of its base is 4 cm. Calculate the total surface area of the article.

A model is shaped like a cylinder with two cones attached to its ends. The total length of the model is 15 cm and its diameter is 4 cm. If each cone has a height of 3 cm, analyze the dimensions and find the volume of the model.

A solid iron pole consists of a cylinder of height 150 cm and base diameter 20 cm, surmounted by another cylinder of height 50 cm and radius 6 cm. Calculate the volume of the pole. (Use $\pi = 3.14$)

A juice glass is cylindrical with an inner diameter of 6 cm. The bottom of the glass has a hemispherical raised portion that reduces its capacity. If the height of the glass is 12 cm, calculate the actual capacity of the glass. (Use $\pi = 3.14$)

Explain the difference between the 'apparent capacity' and 'actual capacity' of a cylindrical glass that has a hemispherical raised portion at the bottom.

A solid is formed by mounting a cone on a hemisphere of the same radius. Explain which surface areas are included in the total surface area of this new solid.

Summarize the procedure to find the volume of air in a shed shaped like a cuboid surmounted by a half-cylinder, after accounting for machinery and workers inside.

Identify the component areas needed to calculate the total surface area of a bird-bath shaped like a cylinder with a hemispherical depression at one end.

Describe the steps to find the height of the conical part of a toy that is composed of a cone mounted on a hemisphere, given the total height of the toy and the radius of the hemisphere.

Two cubes, each with a volume of 125 cm³, are joined end to end. Calculate the surface area of the resulting cuboid.

A solid is composed of a cylinder with hemispherical ends. If the total length of the solid is 28 cm and the diameter of the cylinder is 10 cm, calculate its volume. (Use $\pi = \frac{22}{7}$)

Explain in detail the general principle for finding the surface area of a combination of solids. Use the example of a decorative block made of a cube with a hemisphere fixed on top to illustrate your explanation. Specifically, mention why an area is subtracted.

Describe the method to find the volume of a solid formed by joining two basic solids. Contrast this with the method for finding the surface area of the same combined solid. Use the example of a toy shaped like a cone on a hemisphere to illustrate.

Calculate the surface area of a toy which is in the form of a cone of radius 6 cm mounted on a hemisphere of the same radius. The total height of the toy is 14 cm. (Use $\pi = \frac{22}{7}$)

A tent is in the shape of a cylinder surmounted by a cone. The height of the cylindrical part is 3 m and its diameter is 14 m. The slant height of the conical top is 9 m. Calculate the area of the canvas required for the tent, and find the cost if the canvas costs ₹450 per $\text{m}^2$.

A cubical block of side 10 cm is surmounted by a hemisphere. What is the largest diameter the hemisphere can have? Calculate the surface area of the solid with this hemisphere.

Calculate the volume of the largest right circular cone that can be cut out of a cube whose edge is 9 cm.

You have a fixed amount of metal sheet to create a closed cylindrical container. Evaluate which of these two conflicting goals is the practical objective in engineering design: maximizing the container's volume or maximizing its surface area.

Formulate a single algebraic expression for the surface area of a solid composed of a cube of edge '$a$' with a hemispherical depression of diameter '$a$' carved out from one of its faces.

A solid toy is a cylinder of height 8 cm and radius 3 cm, with a conical cavity of the same height and radius hollowed out from it. Another toy is a solid hemisphere of radius 4.1 cm. Evaluate which toy is heavier, assuming both are made of the same wood with uniform density. (Use $\pi = 3.14$)

Create a problem involving a composite solid made from a cylinder and at least one other shape, where the total volume is exactly $300\pi \text{ cm}^3$. Provide the dimensions you chose and verify the volume.

To find the volume of wood in a cubical block of side 10 cm with a hemispherical scoop of diameter 6 cm removed, a student calculates: Volume = (Volume of Cube) + (Volume of Hemisphere). Critique the student's formula and provide the correct calculation. (Use $\pi = 3.14$)

An artist wants to create a solid sculpture by joining two identical solid cones at their bases. The resulting solid is a bicone. The artist claims that the surface area of this new solid is double the total surface area of a single cone. Justify whether this claim is correct.

Design a storage silo in the shape of a cylinder surmounted by a cone. The radius of the base is to be 3 m. The total height of the silo must be 13 m. The cost of material for the cylindrical part is ₹100 per m² and for the conical part is ₹250 per m² (due to complexity). Propose a design by choosing a height for the cylindrical part, and then justify your design by calculating the total material cost. Propose a different height for the cylinder and evaluate if it would be cheaper or more expensive.

A cone of height 24 cm has a curved surface area of $550 \text{ cm}^2$. Calculate the radius of its base. (Use $\pi = \frac{22}{7}$)

A rocket is shaped like a cylinder of radius $R$ and height $H$, surmounted by a cone of the same radius $R$ and height $h$. Formulate a general expression for the ratio of the rocket's total volume to its total surface area (excluding the base). Then, evaluate this "efficiency ratio" for a rocket where $R=2$ m, $H=10$ m, and $h=3$ m.

A cylindrical container with radius 6 cm and height 15 cm is full of ice cream. The ice cream is to be filled into cones of height 12 cm and radius 3 cm, having a hemispherical top. Evaluate how many such cones can be filled with ice cream.

From a solid wooden cube of side 14 cm, a hemisphere of the largest possible diameter is carved out from one face. Calculate the surface area of the remaining solid.

Summarize how to calculate the area to be painted for a wooden toy rocket shaped like a cone mounted on a cylinder, where the base of the cone is wider than the base of the cylinder. List all the individual areas that must be calculated for both the conical (orange) and cylindrical (yellow) portions as described in the source text.

A tent is shaped like a cylinder surmounted by a cone. The radius of the base is $r$. The height of the cylinder is $h_1$ and the height of the cone is $h_2$. Formulate a single expression for the total cost of the canvas required at a rate of ₹C per m², assuming the floor is not covered.

A company wants to create a bird bath from a single solid block of wood shaped like a cylinder with height 50 cm and radius 20 cm. The design requires scooping out a hemisphere from the top face. To save material, the company wants the volume of the wood scooped out to be exactly 25% of the original cylinder's volume. Create the specifications for this bird bath by determining the required radius of the hemispherical depression. Justify whether such a design is physically possible within the dimensions of the block.

Propose a modification to the shape of a standard right circular cylinder that would increase its volume without changing its height or the area of its curved surface.

Given a solid cylinder from which a conical cavity of the same height and same base radius is hollowed out, describe which surfaces constitute the total surface area of the remaining solid.