NCERT Class 10 Maths Solutions: Surface Areas and Volumes

Question:

The problem requires calculating the total surface area of a composite solid formed by a cone mounted on a hemisphere. The key concepts involved are:
1. Calculating the surface area of a hemisphere.
2. Calculating the curved surface area of a cone.
3. Understanding that the base of the cone and the curved surface of the hemisphere are joined, so they are not part of the total surface area.
4. Using the formula for the slant height of a cone.
5. Applying the given value of pi.


The toy is a combination of a cone and a hemisphere.
The radius of the hemisphere and the cone is given as r = 3.5 cm.
The total height of the toy is 15.5 cm.

The total height of the toy is the sum of the height of the cone and the height of the hemisphere.
Height of the hemisphere = radius of the hemisphere = r = 3.5 cm.
Height of the cone (h) = Total height of the toy – Height of the hemisphere
h = 15.5 cm – 3.5 cm = 12 cm.

To find the curved surface area of the cone, we need the slant height (l).
The slant height can be calculated using the Pythagorean theorem: l = sqrt(r^2 + h^2).
l = sqrt((3.5)^2 + (12)^2)
l = sqrt(12.25 + 144)
l = sqrt(156.25)
l = 12.5 cm.

The total surface area of the toy is the sum of the curved surface area of the cone and the surface area of the hemisphere.
Curved surface area of the cone = πrl
Surface area of the hemisphere = 2πr^2

Total surface area of the toy = πrl + 2πr^2
Total surface area = πr(l + 2r)

Substitute the values: r = 3.5 cm, l = 12.5 cm, and π = 22/7.
Total surface area = (22/7) * 3.5 * (12.5 + 2 * 3.5)
Total surface area = (22/7) * 3.5 * (12.5 + 7)
Total surface area = (22/7) * 3.5 * (19.5)

Calculate the value:
Total surface area = 22 * (3.5/7) * 19.5
Total surface area = 22 * 0.5 * 19.5
Total surface area = 11 * 19.5
Total surface area = 214.5 cm^2.

Question:

The problem requires calculating the volume of a composite shape made of a cylinder and a sphere. The total volume is the sum of the volume of the cylindrical neck and the volume of the spherical part. The formula for the volume of a cylinder is V_cylinder = π * r^2 * h, and the formula for the volume of a sphere is V_sphere = (4/3) * π * r^3.


The vessel consists of two parts: a cylindrical neck and a spherical body.
First, calculate the volume of the cylindrical neck.
The length of the cylindrical neck is given as h_cylinder = 8 cm.
The diameter of the cylindrical neck is 2 cm, so the radius is r_cylinder = 2/2 = 1 cm.
The volume of the cylinder is V_cylinder = π * r_cylinder^2 * h_cylinder.
Substitute the values: V_cylinder = 3.14 * (1 cm)^2 * 8 cm = 3.14 * 1 cm^2 * 8 cm = 25.12 cm^3.

Next, calculate the volume of the spherical part.
The diameter of the spherical part is 8.5 cm, so the radius is r_sphere = 8.5/2 = 4.25 cm.
The volume of the sphere is V_sphere = (4/3) * π * r_sphere^3.
Substitute the values: V_sphere = (4/3) * 3.14 * (4.25 cm)^3.
Calculate (4.25 cm)^3 = 4.25 * 4.25 * 4.25 = 76.765625 cm^3.
Now, calculate V_sphere = (4/3) * 3.14 * 76.765625 cm^3.
V_sphere = 1.3333… * 3.14 * 76.765625 cm^3.
V_sphere ≈ 4.1866… * 76.765625 cm^3.
V_sphere ≈ 321.55 cm^3.

Finally, calculate the total volume of the vessel by adding the volume of the cylindrical neck and the volume of the spherical part.
Total Volume = V_cylinder + V_sphere.
Total Volume = 25.12 cm^3 + 321.55 cm^3.
Total Volume = 346.67 cm^3.

The child found the volume to be 345 cm^3.
Comparing the calculated volume with the child’s measurement:
346.67 cm^3 ≠ 345 cm^3.
Therefore, the child is not correct.

The final answer is $\boxed{The child is not correct because the calculated volume is 346.67 cm^3}$.

Question:

The key concepts are: understanding the properties of a cube (all sides equal), calculating the side length of a cube from its volume, understanding how joining two cubes end to end forms a cuboid, and calculating the surface area of a cuboid.


Step 1: Find the side length of one cube.
The volume of a cube is given by the formula V = s^3, where s is the side length.
Given that the volume of each cube is 64 cm^3.
So, s^3 = 64 cm^3.
To find s, we need to find the cube root of 64.
s = ∛64 cm = 4 cm.
Each cube has a side length of 4 cm.

Step 2: Determine the dimensions of the resulting cuboid.
When two identical cubes are joined end to end, the length of the resulting cuboid increases, while the width and height remain the same as the side length of a single cube.
Let the dimensions of the resulting cuboid be length (l), width (w), and height (h).
The original cubes have dimensions 4 cm × 4 cm × 4 cm.
When joined end to end, for example, along their length:
The new length (l) will be the sum of the lengths of the two cubes: l = 4 cm + 4 cm = 8 cm.
The width (w) will be the same as the width of a cube: w = 4 cm.
The height (h) will be the same as the height of a cube: h = 4 cm.
So, the resulting cuboid has dimensions 8 cm × 4 cm × 4 cm.

Step 3: Calculate the surface area of the resulting cuboid.
The surface area of a cuboid is given by the formula SA = 2(lw + lh + wh).
Substitute the dimensions of the cuboid into the formula:
SA = 2 × [(8 cm × 4 cm) + (8 cm × 4 cm) + (4 cm × 4 cm)]
SA = 2 × [32 cm^2 + 32 cm^2 + 16 cm^2]
SA = 2 × [80 cm^2]
SA = 160 cm^2.

The surface area of the resulting cuboid is 160 cm^2.

Question:

The problem involves calculating the volume of two combined cylinders and then using the density to find the mass. The key concepts are the formula for the volume of a cylinder and the relationship between mass, density, and volume (mass = density × volume).


The solid iron pole is composed of two cylinders. We need to calculate the volume of each cylinder separately and then add them to find the total volume of the pole.

Cylinder 1:
Height (h1) = 220 cm
Base diameter = 24 cm, so the radius (r1) = 24/2 = 12 cm.
The formula for the volume of a cylinder is V = πr²h.
Volume of Cylinder 1 (V1) = π * (r1)² * h1
V1 = 3.14 * (12 cm)² * 220 cm
V1 = 3.14 * 144 cm² * 220 cm
V1 = 3.14 * 31680 cm³
V1 = 99475.2 cm³

Cylinder 2:
Height (h2) = 60 cm
Radius (r2) = 8 cm
Volume of Cylinder 2 (V2) = π * (r2)² * h2
V2 = 3.14 * (8 cm)² * 60 cm
V2 = 3.14 * 64 cm² * 60 cm
V2 = 3.14 * 3840 cm³
V2 = 12057.6 cm³

Total Volume of the Pole (V_total) = V1 + V2
V_total = 99475.2 cm³ + 12057.6 cm³
V_total = 111532.8 cm³

Mass of the pole:
Given that 1 cm³ of iron has a mass of 8 g.
Mass = Density × Volume
Mass = 8 g/cm³ × 111532.8 cm³
Mass = 892262.4 g

To convert grams to kilograms, divide by 1000.
Mass in kg = 892262.4 g / 1000 g/kg
Mass in kg = 892.2624 kg

The mass of the pole is 892262.4 g or 892.2624 kg.

Question:

This question requires knowledge of the volume formulas for a cone and a hemisphere. Specifically, the volume of a cone is given by V_cone = (1/3)πr²h and the volume of a hemisphere is given by V_hemisphere = (2/3)πr³. The total volume of the solid will be the sum of these two volumes.


The solid is composed of two parts: a cone and a hemisphere.
The problem states that the radius of the cone and the hemisphere are equal, and this radius is 1 cm. So, r = 1 cm.
The height of the cone is equal to its radius, so h_cone = r = 1 cm.

First, calculate the volume of the cone:
V_cone = (1/3)πr²h_cone
V_cone = (1/3)π(1 cm)²(1 cm)
V_cone = (1/3)π(1 cm³)
V_cone = (1/3)π cm³

Next, calculate the volume of the hemisphere:
V_hemisphere = (2/3)πr³
V_hemisphere = (2/3)π(1 cm)³
V_hemisphere = (2/3)π(1 cm³)
V_hemisphere = (2/3)π cm³

Finally, find the total volume of the solid by adding the volume of the cone and the volume of the hemisphere:
V_solid = V_cone + V_hemisphere
V_solid = (1/3)π cm³ + (2/3)π cm³
V_solid = (1/3 + 2/3)π cm³
V_solid = (3/3)π cm³
V_solid = π cm³

The volume of the solid is π cm³.