NCERT Exemplar Solutions - Class 10 - Mathematics - Unit 12: Surface Areas & Volumes

Exercise 12.2

Step 1: A hemisphere means "half of a sphere". Its surface has two parts:

• The curved surface area (CSA) = \(2\pi r^2\).
• The flat circular base area = \(\pi r^2\).

Step 2: When we take two identical hemispheres and join them at their bases, they form a complete sphere. But here the question says both curved parts and both bases are included in the surface area.

Step 3: Total surface area of the combination = curved surface of both hemispheres + base of both hemispheres.

• Curved surface of two hemispheres = \(2 \times 2\pi r^2 = 4\pi r^2\).
• Base area of two hemispheres = \(2 \times \pi r^2 = 2\pi r^2\).

Step 4: Add them together:

\(4\pi r^2 + 2\pi r^2 = 6\pi r^2\).

Step 5: Since the given total surface area is also \(6\pi r^2\), the statement is true.

Note: Here radius \(r\) is measured in metres (SI unit), so the surface area is in square metres (m²).

Step 1: Understand the situation.

We have two identical cylinders. Each has radius \(r\) (in metres) and height \(h\) (in metres). When one is placed on top of the other, we get a single tall cylinder of:

Height = \(h + h = 2h\) (in metres).

Radius = \(r\) (remains the same, in metres).

Step 2: Recall the formula for total surface area (TSA) of a cylinder:

\[ \text{TSA} = 2\pi r h + 2\pi r^2 \]

Here:

• \(2\pi r h\) is the curved surface area (CSA), measured in square metres (m²).
• \(2\pi r^2\) is the area of the two circular ends (top and bottom), also in m².

Step 3: Apply the formula to the new cylinder.

For height = \(2h\), radius = \(r\):

\[ \text{TSA} = 2\pi r(2h) + 2\pi r^2 \]

Simplify:

\[ \text{TSA} = 4\pi r h + 2\pi r^2 \]

Step 4: Compare with the given statement.

The question says TSA = \(4\pi rh + 4\pi r^2\).

But we found TSA = \(4\pi rh + 2\pi r^2\).

Final Answer: The given statement is incorrect. So the answer is False.

Step 1: Write down the dimensions in SI units.

• Radius of both cone and cylinder = \(r\) metres (m).
• Height of cone and cylinder = \(h\) metres (m).

Step 2: Break the solid into parts.

• One cone (on top).
• One cylinder (at bottom).

Step 3: Formula for curved surface area (CSA).

• CSA of cone = \(\pi r l\), where \(l = \sqrt{r^2+h^2}\) (slant height in metres).
• CSA of cylinder = \(2\pi r h\).

Step 4: Consider the bases.

• The top base of cylinder is covered by the cone, so it is not visible.
• We only need the bottom base of cylinder (circle of radius \(r\)).
• Area of bottom base = \(\pi r^2\).

Step 5: Add them up to get total surface area (TSA).

TSA = CSA of cone + CSA of cylinder + base area of cylinder

= \(\pi r l + 2\pi r h + \pi r^2\) (square metres).

Step 6: Compare with given expression.

Given = \(\pi r[\sqrt{r^2+h^2}+3r+2h]\).

Our result = \(\pi r[\sqrt{r^2+h^2} + 2h + r]\).

These are not the same. The given expression has extra terms.

Final Answer: The statement is false.

Step 1: The ball fits exactly inside the cube. This means the diameter of the ball is equal to the side of the cube. So, diameter of ball = \(a\) (in metres, SI unit).

Step 2: Radius is half of diameter. Therefore, radius \(r = \dfrac{a}{2}\, \text{m}\).

Step 3: Formula for volume of a sphere (ball) is: \[ V = \dfrac{4}{3} \pi r^3 \]

Step 4: Substitute \(r = a/2\): \[ V = \dfrac{4}{3} \pi \left(\dfrac{a}{2}\right)^3 = \dfrac{4}{3} \pi \cdot \dfrac{a^3}{8} = \dfrac{\pi a^3}{6} \]

Step 5: The question states that the volume is \(\dfrac{4}{3}\pi a^3\). But we found the actual volume is \(\dfrac{\pi a^3}{6}\).

Final Answer: Since the given volume is incorrect, the statement is False.

Step 1: Recall the correct formula for the volume of a frustum of a cone.

It is: \(V = \dfrac{1}{3}\pi h (r_1^2 + r_2^2 + r_1r_2)\)

• \(V\) = volume in cubic metres (m³)
• \(h\) = vertical height in metres (m)
• \(r_1, r_2\) = radii of the circular ends in metres (m)

Step 2: Compare the given formula with the correct one.

Given: \(\dfrac{1}{3}\pi h(r_1^2 + r_2^2 - r_1r_2)\)

Correct: \(\dfrac{1}{3}\pi h(r_1^2 + r_2^2 + r_1r_2)\)

Step 3: Notice the difference.

The given formula has a minus sign (\(-r_1r_2\)) instead of a plus sign (\(+r_1r_2\)).

Step 4: Therefore, the statement is wrong.

Final Answer: False

Step 1: The vessel has two parts:

• A cylinder of radius \(r\) and height \(h\).
• At the bottom, a hemispherical portion (half of a sphere) is raised inside the cylinder.

Step 2: Formula for volume of a cylinder:

\(V_{cylinder} = \pi r^2 h\) (in cubic metres if \(r,h\) are in metres).

Step 3: Formula for volume of a sphere:

\(V_{sphere} = \dfrac{4}{3} \pi r^3\).

Step 4: Since the bottom is a hemisphere (half of a sphere), its volume is:

\(V_{hemisphere} = \dfrac{1}{2} \times V_{sphere} = \dfrac{1}{2} \times \dfrac{4}{3}\pi r^3 = \dfrac{2}{3}\pi r^3\).

Step 5: Capacity of vessel = Volume of cylinder − Volume of hemisphere (because the hemisphere takes up space inside the cylinder).

So, \(V = \pi r^2 h - \dfrac{2}{3} \pi r^3\).

Step 6: Take \(\pi r^2\) common:

\(V = \pi r^2 \left(h - \dfrac{2}{3}r\right)\).

Step 7: Write in fraction form:

\(V = \dfrac{\pi r^2}{3} (3h - 2r)\).

Step 8: This matches exactly the given formula. ✅

Hence, the statement is True.

Step 1: A frustum of a cone is formed when a small cone is cut off from a bigger cone by a plane parallel to its base.

Step 2: The formula for the curved surface area (CSA) of a frustum is:

\[ CSA = \pi l (r_1 + r_2) \]

where:

• \(r_1\) = radius of the larger circular base,
• \(r_2\) = radius of the smaller circular base,
• \(l\) = slant height of the frustum.

Step 3: The slant height \(l\) is found using Pythagoras theorem in the right-angled triangle formed by height \(h\), difference of radii \((r_1 - r_2)\), and slant height \(l\).

So,

\[ l = \sqrt{h^2 + (r_1 - r_2)^2} \]

Step 4: But in the given statement, \(l\) was written as:

\[ l = \sqrt{h^2 + (r_1 + r_2)^2} \]

This is wrong, because we must use the difference of radii \((r_1 - r_2)\), not the sum.

Step 5: Therefore, the statement is False.

Step 1: The bucket is said to be open. This means there is no metallic sheet used at the top, so we do not count the top circular area.

Step 2: The frustum (truncated cone) has a curved surface area. This part of the metal sheet is used, so we must include it.

Step 3: The frustum is fixed onto a hollow cylindrical base. The cylinder also has a curved surface area, and that part is also made of metal, so it must be included.

Step 4: The bucket has a bottom circular base (the base of the cylinder). This also requires metal, so its area must be added.

Step 5: Total metallic sheet used = Curved Surface Area of frustum + Curved Surface Area of cylinder + Area of bottom circular base.

Step 6: This matches exactly with the statement in the question.

✅ Therefore, the statement is true.

Note: All areas are measured in square metres (m²) in SI units.