R.d Sharma 2022 Mcqs Solutions for Class 9 Maths Chapter 21 - Surface Area And Volume Of A Sphere

Answer:

A sphere has only a single face. Since there are no sides of a sphere, it has a single continuous face.

Therefore, the correct option is (a)

Answer:

The curved surface area of a hemisphere of radius r is. So, the total surface area of a hemisphere will be the sum of the curved surface area and the area of the base.

Total surface area of a hemisphere of radius r =

Therefore, the correct option is (c)

Answer:

In the given question,

The total surface area of a sphere (S₁) =

The total surface area of a hemisphere (S₂) =

So the ratio of the total surface area of a sphere and a hemisphere will be,

Therefore, the ratio of the surface areas is. So, the correct option is (d)

Answer:

In the given problem, we have a sphere and a cube of equal heights. So, let the diameter of the sphere and side of the cube be x units.

So, volume of the sphere (V₁) =

Volume of the cube (V₂) =

So, to find the ratio of the volumes,

Therefore, the ratio of the volumes of sphere and cube of equal heights is . So, the correct option is (d).

Answer:

In the given problem, the largest sphere is carved out of a cube and we have to find the volume of the sphere.

Side of a cube = 6 cm

So, for the largest sphere in a cube, the diameter of the sphere will be equal to side of the cube.

Therefore, diameter of the sphere = 6 cm

Radius of the sphere = 3 cm

Now, the volume of the sphere =

Therefore, the volume of the largest sphere inside the given cube is . So, the correct option is (b).

Answer:

In the given problem, we have a cylindrical rod of the given dimensions:

Radius of the base (r_c) = x units

Height of the cylinder (h) = 8x units

So, the volume of the cylinder (V_c) =

Now, this cylinder is remolded into spherical balls of same radius. So let us take the number of balls be y.

Total volume of y spheres (V_s) =

So, the volume of the cylinder will be equal to the total volume of y number of balls.

We get,

Therefore, the number of balls that will be made is. So, the correct option is (c)

Answer:

Here, we are given that the ratio of the two spheres of ratio 1:8

Let us take,

The radius of 1^st sphere = r₁

The radius of 1^st sphere = r₂

So,

Volume of 1^st sphere (V₁) =

Volume of 2^nd sphere (V₂) =

Now,

Now, let us find the surface areas of the two spheres

Surface area of 1^st sphere (S₁) =

Surface area of 2^nd sphere (S₂) =

So, Ratio of the surface areas,

Using (1), we get,

Therefore, the ratio of the spheres is. So, the correct option is (b)

Answer:

In the given problem, we are given a cone, a hemisphere and a cylinder which stand on equal bases and have equal heights. We need to find the ratio of their volumes.

So,

Let the radius of the cone, cylinder and hemisphere be x cm.

Now, the height of the hemisphere is equal to the radius of the hemisphere. So, the height of the cone and the cylinder will also be equal to the radius.

Therefore, the height of the cone, hemisphere and cylinder = x cm

Now, the next step is to find the volumes of each of these.

Volume of a cone (V₁) =

Volume of a hemisphere (V₂) =

Volume of a cylinder (V₃) =

So, now the ratio of their volumes = (V₁) : (V₂) : (V₃)

Therefore, the ratio of the volumes of the given cone, hemisphere and the cylinder is . So, the correct option is (a).

Answer:

In the given problem, Let the radius of smaller spherical balls which can be made from a bigger ball be x units.

Here,

The radius of the bigger ball (r₁) = 10 cm

The radius of the smaller ball (r₂) = x cm

The number of smaller balls = 8

So, volume of the big ball is equal to the volume of 8 small balls.

Volume of the big balls = volume of the 8 small balls

Further, solving for x, we get,

Now, surface area of a small ball of radius 5 cm =

Therefore, the surface area of the small spherical ball is. So, the correct option is (a).

Answer:

In the given problem, we are given a sphere inscribed in a cube. So, here we need to find the ratio between the volume of a sphere and volume of a cube. This means that the diameter of the sphere will be equal to the side of the cube. Let us take the diameter as d.

Here,

Volume of a sphere (V₁) =

Volume of a cube (V₂) =

Now, the ratio of the volume of sphere to the volume of the cube =

So, the ratio of the volume of sphere to the volume of the cube is . Therefore, the correct option is (d)

Answer:

In the given problem, we have a solid sphere which is remolded into a solid cone such that the radius of the sphere is equal to the height of the cone. We need to find the radius of the base of the cone.

Here, radius of the solid sphere (r_s) = r cm

Height of the solid cone (h) = r cm

Let the radius of the base of cone (r_c) = x cm

So, the volume of cone will be equal to the volume of the solid sphere.

Therefore, we get,

Therefore, radius of the base of the cone is. So, the correct option is (a).

Answer:

In the given problem, we have a sphere inscribed in a cylinder such that it touches the top, base and the lateral surface of the cylinder. This means that the height and the diameter of the cylinder are equal to the diameter of the sphere.

So, if the radius of the sphere = r

The radius of the cylinder (r_c)= r

The height of the cylinder (h) = 2r

Therefore, Volume of the cylinder =

So, the volume of the cylinder is. Therefore, the correct option is (c).

Answer:

In the given problem, we need to find the ratio between the volume of a sphere and volume of a circumscribing right circular cylinder. This means that the diameter of the sphere and the cylinder are the same. Let us take the diameter as d.

Here,

Volume of a sphere (V₁) =

As the cylinder is circumscribing the height of the cylinder will also be equal to the height of the sphere. So,

Volume of a cylinder (V₂) =

Now, the ratio of the volume of sphere to the volume of the cylinder =

So, the ratio of the volume of sphere to the volume of the cylinder is . Therefore, the correct option is (c)

Answer:

In the given problem, we are given a cone and a hemisphere which have equal bases and have equal volumes. We need to find the ratio of their heights.

So,

Let the radius of the cone and hemisphere be x cm.

Also, height of the hemisphere is equal to the radius of the hemisphere.

Now, let the height of the cone = h cm

So, the ratio of the height of cone to the height of the hemisphere =

Here, Volume of the hemisphere = volume of the cone

Therefore, the ratio of the heights of the cone and the hemisphere is. So, the correct option is (b)

Answer:

Statement-2 (Reason): The volumes V₁, V₂ and surface areas of two spheres are connected by the relation ${\left(\frac{S_1}{S_2}\right)}^3={\left(\frac{V_1}{V_2}\right)}^2$.

Let r₁ and r₂ be the radius of the two circles with surface areas S₁ and S₂ respectively and volumes V₁ and V2 respectively.

So,

$$\begin{gathered} \frac{V_1}{V_2}=\frac{\frac{4}{3}\mathrm{\pi }{r_1}^3}{\frac{4}{3}\mathrm{\pi }{r_2}^3} \\ \Rightarrow \frac{V_1}{V_2}=\frac{{r_1}^3}{{r_2}^3} \\ \Rightarrow \frac{V_1}{V_2}={\left(\frac{r_1}{r_2}\right)}^3 \\ \Rightarrow {\left(\frac{V_1}{V_2}\right)}^{\frac{1}{3}}=\frac{r_1}{r_2} .....\left(1\right) \end{gathered}$$

Now,
$$\begin{gathered} \frac{S_1}{S_2}=\frac{4\mathrm{\pi }{r_1}^2}{4\mathrm{\pi }{{\mathrm{r}}_2}^2} \\ \Rightarrow \frac{S_1}{S_2}=\frac{{r_1}^2}{{r_2}^2} \\ \Rightarrow \frac{S_1}{S_2}={\left(\frac{r_1}{r_2}\right)}^2 \\ \Rightarrow \frac{S_1}{S_2}={\left[{\left(\frac{V_1}{V_2}\right)}^{\frac{1}{3}}\right]}^2 \left[\mathrm{From} \left(1\right)\right] \\ \Rightarrow \frac{S_1}{S_2}={\left(\frac{V_1}{V_2}\right)}^{\frac{2}{3}} \\ \Rightarrow {\left(\frac{S_1}{S_2}\right)}^3={\left(\frac{V_1}{V_2}\right)}^2 \end{gathered}$$

Thus, Statement-2 is true.

Statement-1 (Assertion): If the ratio of the surface areas of two spheres is 4 : 9, then the ratio of their volumes is 8: 27.

Let S₁ and S₂ be the surface areas of two spheres respectively and volumes V₁ and V₂ be the volumes of two spheres respectively respectively.

Here,  S₁ : S₂ = 4 : 9

Now, according to Statement-2 
$$\begin{gathered} {\left(\frac{S_1}{S_2}\right)}^3={\left(\frac{V_1}{V_2}\right)}^2 \\ \Rightarrow {\left(\frac{4}{9}\right)}^3={\left(\frac{V_1}{V_2}\right)}^2 \\ \Rightarrow \frac{64}{729}={\left(\frac{V_1}{V_2}\right)}^2 \\ \Rightarrow \sqrt{\frac{64}{729}}=\frac{V_1}{V_2} \\ \Rightarrow \frac{V_1}{V_2}=\frac{8}{27} \end{gathered}$$
⇒ V₁ : V₂ = 8 : 27
Thus, Statement-1 is true.

So, Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Hence, the correct answer is option (a).

Answer:

Statement-2 (Reason): If the volumes of two spheres are in the ratio V₁: V₂, then their radii are in the ratio $V_1^{\frac{1}{3}}:V_2^{\frac{1}{3}}$.
Let r₁ and r₂ be the radius of the two spheres with volumes V₁ and V₂ respectively.
$$\begin{gathered} \frac{V_1}{V_2}=\frac{\frac{4}{3}\mathrm{\pi }{r_1}^3}{\frac{4}{3}\mathrm{\pi }{r_2}^3} \\ \Rightarrow \frac{V_1}{V_2}=\frac{{r_1}^3}{{r_2}^3} \\ \Rightarrow \frac{V_1}{V_2}={\left(\frac{r_1}{r_2}\right)}^3 \\ \Rightarrow {\left(\frac{V_1}{V_2}\right)}^{\frac{1}{3}}=\frac{r_1}{r_2} \\ \Rightarrow \frac{r_1}{r_2}=\frac{{V_1}^{{\displaystyle \frac{1}{3}}}}{{V_2}^{\frac{1}{3}}} \end{gathered}$$

Thus, Statement-2 is true.

Statement-1 (Assertion): If the volumes of two spheres are in the ratio 27: 125, then their radii are in the ratio 3 : 5.
Let r₁ and r₂ be the radius of the two spheres with volumes V₁ and V₂ respectively.

Here,  V₁ : V₂ = 27 : 125

Now, according to Statement-2 
$$\begin{gathered} \frac{{V_1}^{{\displaystyle \frac{1}{3}}}}{{V_2}^{\frac{1}{3}}}=\frac{r_1}{r_2} \\ \Rightarrow {\left(\frac{V_1}{V_2}\right)}^{\frac{1}{3}}=\frac{r_1}{r_2} \\ \Rightarrow {\left(\frac{27}{125}\right)}^{\frac{1}{3}}=\frac{r_1}{r_2} \\ \Rightarrow {\left(\frac{3^3}{5^3}\right)}^{\frac{1}{3}}=\frac{r_1}{r_2} \\ \Rightarrow \frac{3}{5}=\frac{r_1}{r_2} \end{gathered}$$

⇒ r₁ : r₂ = 3 : 5
Thus, Statement-1 is true.

So, Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Hence, the correct answer is option (a).

Answer:

Statement-2 (Reason): If a hemisphere and a cylinder stand on equal bases and have the same height, then their volumes are in the ratio 3 : 2.

Let r and h be the radius and height of the hemisphere and the cylinder standing on equal bases.
Since the hemisphere and the cylinder have the same height so the height of the cylinder is equal to the radius of the hemisphere.
⇒ h = r                 .....(1)
$\begin{array}{rcl}\frac{\mathrm{Volume} \mathrm{of} \mathrm{hemisphere}}{\mathrm{Volume} \mathrm{of} \mathrm{cylinder}}& =& \frac{{\displaystyle \frac{2}{3}}\mathrm{\pi }{r}^3}{\mathrm{\pi }{r}^{2}h}\\ & =& \frac{{\displaystyle \frac{2}{3}r}}{r} \left[\mathrm{From} \left(1\right)\right]\\ & =& \frac{2}{3}\end{array}$

Thus, Statement-2 is false.

Statement-1 (Assertion): The volume of a sphere is equal to two-third of the volume of a cylinder whose height and diameter of the base are equal to the diameter of the sphere.

Let r and h be the radius and height of the sphere and the cylinder whose height and diameter of the base are equal to the diameter of the sphere.
⇒ h = 2r                 .....(1)
Now,

$$\begin{gathered} \frac{\mathrm{Volume} \mathrm{of} \mathrm{sphere}}{\mathrm{Volume} \mathrm{of} \mathrm{cylinder}}=\frac{{\displaystyle \frac{4}{3}}\mathrm{\pi }{r}^3}{\mathrm{\pi }{r}^{2}h} \\ \Rightarrow \frac{\mathrm{Volume} \mathrm{of} \mathrm{sphere}}{\mathrm{Volume} \mathrm{of} \mathrm{cylinder}}=\frac{{\displaystyle \frac{4}{3}r}}{2r} \left[\mathrm{From} \left(1\right)\right] \\ \Rightarrow \frac{\mathrm{Volume} \mathrm{of} \mathrm{sphere}}{\mathrm{Volume} \mathrm{of} \mathrm{cylinder}}=\frac{2}{3} \\ \Rightarrow \mathrm{Volume} \mathrm{of} \mathrm{sphere}=\frac{2}{3}\mathrm{Volume} \mathrm{of} \mathrm{cylinder} \end{gathered}$$

Thus, Statement-1 is true.
So, Statement-1 is true, Statement-2 is false.

Hence, the correct answer is option (c).

Answer:

Statement-2 (Reason): The volume of the largest right circular cone that can be fitted in a cube whose edge is 2r equals to the volume of a hemisphere of radius r.

Given that, the edge of the cube is 2r.
The radius of the largest right circular cone that can be fitted in a cube whose edge is 2r (R)= 2r
The height of the largest right circular cone that can be fitted in a cube whose edge is 2r (H) = 2r
The radius of the hemisphere = r

$\begin{array}{rcl}\mathrm{Volume} \mathrm{of} \mathrm{cone}& =& \frac{1}{3}\mathrm{\pi }{R}^{2}H\\ & =& \frac{1}{3}\mathrm{\pi }{\mathit{\left(}\mathit{2}r\mathit{\right)}}^2\left(2r\right)\\ & =& \frac{1}{3}\mathrm{\pi }\left(8r^3\right)\\ & =& \frac{8}{3}\mathrm{\pi }{r}^3\end{array}$

$\begin{array}{rcl}\mathrm{Volume} \mathrm{of} \mathrm{hemisphere}& =& \frac{4}{3}\mathrm{\pi }{r}^3\\ & & \\ & & \end{array}$

Thus, Statement-2 is false.

Statement-1 (Assertion): If a sphere is inscribed in a cube, then the ratio of the volume of the cube to the volume of the sphere is 6 : π.
Let a be the edge of the cube.
The diameter of the sphere is inscribed in a cube (d) = a
The radius of the sphere is inscribed in a cube (r) =$\frac{a}{2}$
$\begin{array}{rcl}\frac{\mathrm{Volume} \mathrm{of} \mathrm{cube}}{\mathrm{Volume} \mathrm{of} \mathrm{sphere}}& =& \frac{a^3}{\frac{4}{3}\mathrm{\pi }{r}^3}\\ & =& \frac{a^3}{\frac{4}{3}\mathrm{\pi }{\left({\displaystyle \frac{a}{2}}\right)}^3}\\ & =& \frac{3a^3}{4\mathrm{\pi }{{\displaystyle \frac{a}{8}}}^3}\\ & =& \frac{3}{\mathrm{\pi }{\displaystyle \frac{1}{2}}}\\ & =& \frac{6}{\mathrm{\pi }}\\ & & \\ & & \end{array}$

⇒ volume of the cube : volume of the sphere = 6 : π.

Thus, Statement-1 is true.
So, Statement-1 is true, Statement-2 is false.

Hence, the correct answer is option (c).

Answer:

Statement-2 (Reason): If the ratio of radii of two spheres is 2 : 3, then the ratio of their volumes is 8 : 27.

Let r₁ and r₂ be the radius of the two spheres with volumes V₁ and V₂ respectively.
Given that, r₁ : r₂ = 2 : 3
$$\begin{gathered} \frac{V_1}{V_2}=\frac{\frac{4}{3}\mathrm{\pi }{r_1}^3}{\frac{4}{3}\mathrm{\pi }{r_2}^3} \\ \Rightarrow \frac{V_1}{V_2}=\frac{{r_1}^3}{{r_2}^3} \\ \Rightarrow \frac{V_1}{V_2}={\left(\frac{r_1}{r_2}\right)}^3 \\ \Rightarrow \frac{V_1}{V_2}={\left(\frac{2}{3}\right)}^3=\frac{2^3}{3^3} \\ \Rightarrow \frac{V_1}{V_2}=\frac{8}{27} \end{gathered}$$
⇒ V₁ : V₂ = 8 : 27
Thus, Statement-2 is true.

Statement-1 (Assertion): If the ratio of the surface areas of two spheres is 4 : 25, then the ratio of their radii is 4 : 5.
Let S₁ and S₂ be the surface areas of two spheres respectively and  r₁ and r₂ be the radius of two spheres respectively.

Here,  S₁ : S₂ = 4 : 25

Now, according to Statement-2 
$$\begin{gathered} \frac{S_1}{S_2}=\frac{4\mathrm{\pi }{r_1}^2}{4\mathrm{\pi }{{\mathrm{r}}_2}^2} \\ \Rightarrow \frac{S_1}{S_2}=\frac{{r_1}^2}{{r_2}^2} \\ \Rightarrow \frac{4}{25}={\left(\frac{r_1}{r_2}\right)}^2 \\ \Rightarrow \sqrt{\frac{4}{25}}=\frac{r_1}{r_2} \\ \Rightarrow \frac{2}{5}=\frac{r_1}{r_2} \end{gathered}$$
⇒ r₁ : r₂ = 2 : 5
Thus, Statement-1 is false.

So, â€‹Statement-1 is false, Statement-2 is true.

Hence, the correct answer is option (d).