Tag: maths

Questions Related to maths

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 $
= $56972 \space\ cm^3$

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.

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. $103.035 \space\ in^3$

  2. $93.035 \space\ in^3$

  3. $123.035 \space\ in^3$

  4. $113.035 \space\ in^3$

Show answer
Correct Option: D
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 $
= $113.035 \space\ in^3$

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.

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. $8(18)^{1/3}$ cm

  2. $10$ cm

  3. $12$ cm

  4. $18(18)^{1/3}$ cm

Show answer
Correct Option: A
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

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

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

  1. $17m$

  2. $19m$

  3. $18m$

  4. $20m$

Show answer
Correct Option: A
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

Which of the following alphabet represents a closed curve?

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

  1. U

  2. R

  3. C

  4. O

Show answer
Correct Option: D
Explanation:

In simple closed curves, the shapes are closed by line-segments or by a curved line.

Triangle, quadrilateral, circle, etc., are examples of closed curves.

Instruments used to draw circle

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

  1. scale and setsquare

  2. scale and protractor

  3. scale and compass

  4. scale

Show answer
Correct Option: C
Explanation:

In order to draw a circle, first we need a compass and we need to take the required radius length with the help of a scale.

So, the required tools are scale and compass.

When you draw without lifting your pencil or pen on a paper you get?

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

  1. Plane curve

  2. Plane shadow

  3. 3-dimension

  4. Line

Show answer
Correct Option: A
Explanation:

When we join no. of points without lifting our pen we make a shape which not necessarily should be straight or curve. We get a plane curve.
Therefore, A is the correct answer.