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Multiple choice 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

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

  1. $\approx 9000 \space\ m^3$
  2. $\approx 8000 \space\ m^3$
  3. $\approx 10000 \space\ m^3$
  4. $\approx 7000 \space\ m^3$
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $
$\approx 10000 \space\ m^3$

Multiple choice 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

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?

  1. $610.293 \space\ ft^3$
  2. $510.293 \space\ ft^3$
  3. $450.293 \space\ ft^3$
  4. $410.293 \space\ ft^3$
Reveal answer Fill a bubble to check yourself
D Correct answer
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 $
= $410.293 \space\ ft^3$

Multiple choice 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

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.

  1. $257.91 \space\ cm^3$
  2. $457.91 \space\ cm^3$
  3. $417.91 \space\ cm^3$
  4. $357.91 \space\ cm^3$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $
= $457.91 \space\ cm^3$

Multiple choice 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

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

  1. $44972 \space\ cm^3$
  2. $56972 \space\ cm^3$
  3. $66972 \space\ cm^3$
  4. $56000 \space\ cm^3$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $
= $56972 \space\ cm^3$

Multiple choice 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

Find the volume of material that is needed to form a spherical shell whose outer radius is $3.0$ inches and whose inner radius is $0.1$ inches.

  1. $103.035 \space\ in^3$
  2. $93.035 \space\ in^3$
  3. $123.035 \space\ in^3$
  4. $113.035 \space\ in^3$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$R = 3.0$ in
$r  = 0.1$ in
Volume = $\cfrac{4}{3}\pi (R^3-r^3)$
= $\cfrac{4}{3}\pi (3^3-0.1^3)$
= $\cfrac{4}{3}\pi (27-0.001)$
= $\cfrac{4}{3}\pi (26.999)$
= $\cfrac{107.996 \pi}{3}$
= $35.998\pi $
= $113.035 \space\ in^3$

Multiple choice 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

A spherical shell of lead, whose external diameter is $24$ cm, is melted and recast into a right circular cylinder, whose height is $12$ cm and diameter $16$ cm. Determine the internal diameter of the shell.

  1. $8(18)^{1/3}$ cm
  2. $10$ cm
  3. $12$ cm
  4. $18(18)^{1/3}$ cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

  

Outer radius of the spherical lead $ = \dfrac {24}{2} = 12 $ cm 
Radius of the cylinder $ = \dfrac {16}{2} = 8 $ cm  
Since the spherical lead is recasted into the cylinder, their volumes are equal. 
Volume of a hollow sphere of outer radius $R$ and inner radius $r$ $ = \dfrac { 4 }{ 3 } \pi ({R}^{3} -{ r }^{ 3 }) $
Volume of a Cylinder of Radius "$R$" and height "$h$" $ = \pi { R }^{ 2 }h $
Hence, $ \dfrac { 4 }{ 3 } \pi ({12}^{3} -{ r }^{ 3 }) = \pi { 8 }^{ 2 } \times 12 $ 
Thus $ 1728 - { r }^{ 3 } = 576 $
$\Rightarrow  { r }^{ 3 } = 1152 $
$\Rightarrow  r = \sqrt [3] {1152} = 4 \sqrt [3] {18}   $ cm 
Inner diameter of the spherical lead $ = 2 \times \ \text{radius }= 2 \times 4 \sqrt [3] {18} $ cm $= 8 \sqrt [3] {18} $ cm

Multiple choice maths fundamental concepts - geometry terms related to polygons curves open and closed figures

A man goes $15m$ due west and then $8m$ due north. How far is he from the straight point?

  1. $17m$
  2. $19m$
  3. $18m$
  4. $20m$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The least distance between the start and the end point can be calculated by the square sum root of the two distances travelled since north and west directions are perpendicular to each. hence, 
Least distance = $\sqrt{(15^2 + 8^2)}$
Least distance = 17 m