Tag: surface area and volume of sphere

Questions Related to surface area and volume of sphere

A metallic spherical shell of internal and external diameters $8 cm$ and $12 cm$, respectively is melted and recast into the form of a cone of base diameter $8 cm$. The height of the cone is

maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

  1. $114 cm$

  2. $76 cm$

  3. $38 cm$

  4. $19 cm$

Show answer
Correct Option: C
Explanation:

Volume of a hollow sphere of outer Radius R and inner radius r $ =

\cfrac { 4 }{ 3 } \pi ({R}^{3} -{ r }^{ 3 }) $

Inner radius of the spherical shell $ = \cfrac {8}{2} = 4 cm $

Outer radius of the spherical shell $ = \cfrac {12}{2} = 6  cm $

Volume of a cone $ = \cfrac { 1 }{ 3 } \pi { r }^{ 2 }h $  where r is the

radius of the base of the cone and h is the height.
Radius of the cone $ = \cfrac{8}{2} = 4  cm $

Now, Volume of the hollow sphere $ = $ Volume of cone
$ => \cfrac { 4 }{ 3 } \pi ({6}^{3} -{ 4 }^{ 3 }) = \cfrac { 1 }{ 3 } \pi { 4 }^{ 2 }h $
$ => 4 \times(216-64) = 16h $
$ h = 38  cm $

The radius of the smaller circle is $2$ m and the radius of the larger circle is $10$ m. What is the volume of the of the spherical shell inscribed in the larger circle?

maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

  1. $3153.17 \space\ m^3$

  2. $4153.17 \space\ m^3$

  3. $2153.17 \space\ m^3$

  4. $153.17 \space\ m^3$

Show answer
Correct Option: B
Explanation:

$R = 10 m$$
$r  = 2 m$
Volume $=\cfrac{4}{3}\pi (R^3-r^3)$
$=\cfrac{4}{3}\pi (10^3-2^3)$

$=\cfrac{4}{3}\pi (1000-8)$
$=\cfrac{4}{3}\pi (992)$
$=\cfrac{3968 \pi}{3}$
$=1322.66\pi $
$=4153.17 \space\ m^3$

The radius of the smaller circle is 4 cm and the radius of the larger circle is 8 cm. Find the volume of the of the spherical shell inscribed in the larger circle.

maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

  1. $1875.62 \space\ cm^3$

  2. $875.62 \space\ cm^3$

  3. $2875.62 \space\ cm^3$

  4. $3875.62 \space\ cm^3$

Show answer
Correct Option: A
Explanation:

R $= 8\ cm$
r  $= 4\ cm$
Volume = $\dfrac{4}{3}\pi (R^3-r^3)$


= $\dfrac{4}{3}\pi (8^3-4^3)$

= $\dfrac{4}{3}\pi (512-64)$

= $\dfrac{4}{3}\pi (448)$

= $\dfrac{1792 \pi}{3}$
= $1875.62 \space\  cm^3$

The inside radius of a spherical metal shell is $25$ cm and the thickness of the shell is $10$ cm. Calculate the volume of the material used in the shell to the nearest unit.

maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

  1. $124,087 \space\ cm$

  2. $144,087 \space\ cm$

  3. $114,087 \space\ cm$

  4. $134,087 \space\ cm$

Show answer
Correct Option: C
Explanation:

Thickness, $T = R - r$
$10 = R - 25$
$R = 35$ cm
$r  = 25$ cm
Volume = $\cfrac{4}{3}\pi (R^3-r^3)$
$=$ $\cfrac{4}{3}\pi (35^3-25^3)$
$=$ $\cfrac{4}{3}\pi (42875-15625)$
$=$ $\cfrac{4}{3}\pi (27250)$
$=$ $\cfrac{109000 \pi}{3}$
$=$ $114,087 \space\ cm$

A spherical shell 5 m thick has an outer radius of 7 m. What is the volume of shell?

maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

  1. $1202.53\space\ m$

  2. $1302.53\space\ m$

  3. $1402.53\space\ m$

  4. $1102.53\space\ m$

Show answer
Correct Option: C
Explanation:

Thickness, $T = R - r$
$5 = 7 - r$
$r = 7 - 5 = 2$ m
$R = 7$ m
Volume = $\cfrac{4}{3}\pi (R^3-r^3)$
= $\cfrac{4}{3}\pi (7^3-2^3)$
= $\cfrac{4}{3}\pi (343-8)$
= $\cfrac{4}{3}\pi (335)$
= $\cfrac{1340 \pi}{3}$
= $1402.53\space\ m$

A hollow spherical shell has inner diameter $4$ cm and outer diameter $8$ cm. Determine the volume of the shell.

maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

  1. $204.45 \space\ cm^3$

  2. $134.45 \space\ cm^3$

  3. $234.45 \space\ cm^3$

  4. $334.45 \space\ cm^3$

Show answer
Correct Option: C
Explanation:

Outer radius, $R = 4$ cm
Inner radius, $r  = 2$ cm
Volume = $\cfrac{4}{3}\pi (R^3-r^3)$
= $\cfrac{4}{3}\pi (4^3-2^3)$
= $\cfrac{4}{3}\pi (64-8)$
= $\cfrac{4}{3}\pi (56)$
= $\cfrac{224 \pi}{3}$
= $74.666\pi $
= $234.45 \space\ cm^3$

A spherical shell has a outer radius $14$ m and inner radius $7$ m. What's the volume of the sphere?

maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

  1. $\approx 9000 \space\ m^3$

  2. $\approx 8000 \space\ m^3$

  3. $\approx 10000 \space\ m^3$

  4. $\approx 7000 \space\ m^3$

Show answer
Correct Option: C
Explanation:

Outer radius, $R = 14$ cm
Inner radius, $r  = 7$ cm
Volume = $\cfrac{4}{3}\pi (R^3-r^3)$
= $\cfrac{4}{3}\pi (14^3-7^3)$
= $\cfrac{4}{3}\pi (2744-343)$
= $\cfrac{4}{3}\pi (2401)$
= $\cfrac{9604 \pi}{3}$
= $3201.33\pi $
$\approx 10000 \space\ m^3$

What is the volume of material that is needed to form a spherical shell whose outer radius is $5$ ft and whose inner radius is $3$ ft?

maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

  1. $610.293 \space\ ft^3$

  2. $510.293 \space\ ft^3$

  3. $450.293 \space\ ft^3$

  4. $410.293 \space\ ft^3$

Show answer
Correct Option: D
Explanation:

$R = 5$ ft
$r  = 3$ ft
Volume = $\cfrac{4}{3}\pi (R^3-r^3)$
= $\cfrac{4}{3}\pi (5^3-3^3)$
= $\cfrac{4}{3}\pi (125-27)$
= $\cfrac{4}{3}\pi (98)$
= $\cfrac{392 \pi}{3}$
= $130.666\pi $
= $410.293 \space\ ft^3$

Calculate the volume of the material used in the shell to the nearest unit. The inside radius of a spherical metal shell is $2.5$ cm and the outer radius of the shell is $5$ cm.

maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

  1. $257.91 \space\ cm^3$

  2. $457.91 \space\ cm^3$

  3. $417.91 \space\ cm^3$

  4. $357.91 \space\ cm^3$

Show answer
Correct Option: B
Explanation:

$R = 5$ cm
$r  = 2.5$ cm
Volume = $\cfrac{4}{3}\pi (R^3-r^3)$
= $\cfrac{4}{3}\pi (5^3-2.5^3)$
= $\cfrac{4}{3}\pi (125-15.625)$
= $\cfrac{4}{3}\pi (109.375)$
= $\cfrac{437.5 \pi}{3}$
= $1312.5\pi $
= $457.91 \space\ cm^3$

Determine the volume of a spherical shell which has an inner radius of $6$ cm and an outer radius of $24$ cm.

maths solids volume of a sphere surface area and volume of sphere surface areas and volumes surface areas and volumes of solids problems involving volume of combined solids application of surface area and volume of solids

  1. $44972 \space\ cm^3$

  2. $56972 \space\ cm^3$

  3. $66972 \space\ cm^3$

  4. $56000 \space\ cm^3$

Show answer
Correct Option: B
Explanation:

Outer radius, $R = 24$ cm
Inner radius, $r  = 6$ cm
Volume = $\cfrac{4}{3}\pi (R^3-r^3)$
= $\cfrac{4}{3}\pi (24^3-6^3)$
= $\cfrac{4}{3}\pi (13824-216)$
= $\cfrac{4}{3}\pi (13608)$
= $\cfrac{54432 \pi}{3}$
= $18144\pi $
= $56972 \space\ cm^3$