Class 10 Mathematics Chapter 12 Surface Areas And Volumes

Key Concept: Surface Area Calculation

c) 502.4 cm²

[Solution Description] To find the total surface area, we first calculate the curved surface area (CSA) of the cylinder using $CSA = 2\pi rh$. Here, $r = 4$ cm and $h = 12$ cm.

$CSA_{\text{cylinder}} = 2 \times \pi \times 4 \times 12 = 96\pi \, \text{cm}^2$

Next, we calculate the surface area of one hemisphere using $SA = 2\pi r^2$. Since there are two hemispheres, we double this value.

$SA_{\text{hemispheres}} = 2 \times 2\pi \times 4^2 = 64\pi \, \text{cm}^2$

Finally, add the areas:

$Total \, Surface \, Area = 96\pi + 64\pi = 160\pi \, \text{cm}^2$

Approximating $\pi \approx 3.14$, the total surface area is approximately $502.4 \, \text{cm}^2$.

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Key Concept: Application of TSA

c) $125.66 cm^2$
[Solution Description] The radius of the cylinder and hemisphere is $ r = \frac{4}{2} = 2 cm$.

The curved surface area of the cylinder is calculated as:

$
CSA_{\text{cylinder}} = 2\pi rh = 2\pi \times 2 \times 6 = 24\pi
$

The curved surface area of one hemisphere is:

$
CSA_{\text{hemisphere}} = 2\pi r^2 = 2\pi \times (2)^2 = 8\pi
$

Since there are two hemispheres, their combined surface area would be:

$
2 \times CSA_{\text{hemisphere}} = 2 \times 8\pi = 16\pi
$

Therefore, the total surface area (TSA) of the toy is:

$
TSA = CSA_{\text{cylinder}} + 2 \times CSA_{\text{hemisphere}} = 24\pi + 16\pi = 40\pi
$

Converting to numerical value, the TSA is approximately $ 40\pi \approx 125.66  cm^2$.

The total surface area (TSA) of a toy shaped like a cylinder with two hemispheres on its ends is calculated by summing the curved surface area (CSA) of the cylindrical part and the CSAs of the two hemispheres. The TSA is given by the formula: $ \text{TSA} = \text{CSA}_{\text{cylinder}} + 2 \times \text{CSA}_{\text{hemisphere}} $. The CSA of a cylinder is calculated as $ \text{CSA}_{\text{cylinder}} = 2\pi rh $, where r is the radius and h is the height. The CSA of a hemisphere is $ \text{CSA}_{\text{hemisphere}} = 2\pi r^2 $.

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Key Concept: Combined Surface Area

a) $120\pi \, \text{cm}^2$

[Solution Description] The total surface area (TSA) of the solid consists of the curved surface area (CSA) of the cylinder plus the CSA of the hemisphere. Using the formulas:

– CSA of the cylinder: $2\pi rh = 2\pi(5)(7) = 70\pi$

– CSA of the hemisphere: $2\pi r^2 = 2\pi(5)^2 = 50\pi$

Therefore, TSA = CSA of cylinder + CSA of hemisphere:

$\text{TSA} = 70\pi + 50\pi = 120\pi \, \text{cm}^2$

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Key Concept: Surface Area Calculation

b) $68.13 cm^2$
[Solution Description]
The total surface area includes the exposed surfaces of both the cube and the hemisphere.
For the cube:

Surface area of the cube = $6s^2 = 6(3)^2 = 54 cm^2$.

For the hemisphere:

Curved surface area = $2\pi r^2 = 2\pi (1.5)^2 = 4.5\pi cm^2$.

Since the base of the hemisphere is attached to the cube, we do not include it.

Total surface area = Surface area of cube + Curved surface area of hemisphere = $54 + 4.5\pi \approx 68.13 cm^2$.

When calculating the total surface area of a solid formed by combining a cube and a hemisphere, it is important to consider only the exposed surfaces. The total surface area is the sum of the surface area of the cube and the curved surface area of the hemisphere, excluding the base of the hemisphere that is attached to the cube. This approach helps in determining the exact surface area exposed to the environment.

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Key Concept: Direct Formula Application

c) $339.12 \, \text{cm}^2$
[Solution Description] The total surface area (TSA) of the capsule consists of the CSA of the cylinder and the CSA of the two hemispheres.

First, calculate the CSA of the cylinder using $2\pi rh$, where $r = \frac{6}{2} = 3$ cm and h = 12 cm:
$
\text{CSA of cylinder} = 2 \times \pi \times 3 \times 12 = 72\pi \, \text{cm}^2.
$

The CSA of one hemisphere is $2\pi r^2$, so two hemispheres give $4\pi r^2$:
$
\text{CSA of hemispheres} = 4 \times \pi \times 3^2 = 36\pi \, \text{cm}^2.
$

Adding them gives the TSA of the capsule:
$
TSA = 72\pi + 36\pi = 108\pi \, \text{cm}^2.
$
With $\pi \approx 3.14$, TSA is approximately $339.12 \, \text{cm}^2$.

The total surface area of a capsule-shaped object, which consists of a cylinder and two hemispheres, is found by adding the CSA of the cylinder and the CSA of the two hemispheres. The CSA of the cylinder is $2\pi rh,$ and the CSA of the hemispheres is $4\pi r^2$, where r is the radius and h is the height of the cylinder.

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Key Concept: Basic Calculation

b) $400 m^3$
[Solution Description] The volume of a cuboid is given by $V = l \times b \times h$.

Calculation:
$
V = 20 \times 10 \times 2 = 400 \text{ m}^3
$
Thus, the swimming pool can hold $400 m^3$ of water.

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Key Concept: Basic Shapes

d) Cuboid
[Solution Description] A basic solid shape with six rectangular faces is known as a cuboid.

A cuboid is a basic solid shape characterized by having six rectangular faces. It is a three-dimensional figure with opposite faces being equal and parallel, and all angles being right angles. The faces of a cuboid are rectangles, and it has 12 edges, 8 vertices, and 6 faces.

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Key Concept: Basic Calculation

c) $\frac{500}{3}\pi \text{cm}^3$
[Solution Description]
The volume V of a sphere is given by the formula:

$
V = \frac{4}{3}\pi r^3
$

Substituting $r = 5 \, \text{cm}$,

$
V = \frac{4}{3}\pi \times 5^3
$

$
= \frac{4}{3}\pi \times 125
$

$
= \frac{500}{3}\pi \, \text{cm}^3
$

The volume of the sphere is $\frac{500}{3}\pi \, \text{cm}^3$.

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Key Concept: Volume with Constraints

b) $72\pi$ cubic meters

[Solution Description]

To find the remaining volume, subtract the volume of the cone from the volume of the cylinder.

The volume of the cylinder $(V_c)$ is calculated using $V = \pi r^2 h$, where $r = 3$ m and $h = 10$ m:

$V_c = \pi \times (3)^2 \times 10 = 90\pi \text{ cubic meters}$

The volume of the cone $(V_{cone})$ is calculated using $V = \frac{1}{3} \pi r^2 h$, where $r = 3$ m and $h = 6$ m:

$V_{cone} = \frac{1}{3} \pi \times (3)^2 \times 6 = 18\pi \text{ cubic meters}$

Therefore, the volume of the remaining solid is:

$V_{\text{remaining}} = V_c – V_{cone} = 90\pi – 18\pi = 72\pi \text{ cubic meters}$

Thus, the volume of the remaining solid is $72\pi$ cubic meters.

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Key Concept: Basic Formula Application

c) $235.5 cm^3$
[Solution Description] To find the total volume, we need to calculate the volume of the cone and the volume of the cylinder separately and then add them.

Volume of the cone:
$
V_{\text{cone}} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (3)^2 (10) = 30\pi
$

Volume of the cylinder:
$
V_{\text{cylinder}} = \pi r^2 h = \pi (3)^2 (5) = 45\pi
$

Total volume:
$
V_{\text{total}} = 30\pi + 45\pi = 75\pi
$

Approximating using $$\pi \approx 3.14$$,
$
V_{\text{total}} \approx 75 \times 3.14 = 235.5 \text{ cubic cm}
$