Tag: recall the surface areas and volumes of different solid shapes

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Multiple choice maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

The corner of a cube_has  been cut by the plane passing through mid-point of  the three edges meeting at that corner. If the edge of  the cube is of 2 cm length,  then the volume of the  pyramid thus cut off is

  1. $\dfrac{1}{24}cm^3$
  2. $\dfrac{1}{6}cm^3$
  3. $\dfrac{1}{48}cm^3$
  4. $6cm^3$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The base of the pyramid thus out off will be a right angled triangle whose sides containing the right angle will be each equal to 1 cm. The height of the pyramid will also be equal to 1 cm. Hence, the volume will be equal to $\frac{1}{6}  cm^3$.

Multiple choice maths perimeter, area and volume volume of prism and pyramid surface areas and volumes of solids recall the surface areas and volumes of different solid shapes

Each side of the base of a square pyramid is reduced by $20%$. By what percent must the height be increased so that the volume of the new pyramid is the same as the volume of the original pyramid?

  1. 20

  2. 40

  3. 46.875

  4. 56.25

  5. 71.875
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Let $a$ be the side of the square.
Length of side of square when reduced by $20\% = a-\dfrac{20a}{100}=0.8a$
Let $a _1=0.8a$
Volume of pyramid $V=\dfrac { 1 }{ 3 } \times $ Area of base $\times height=\dfrac{1}{3}A\times h$
Area of base with side $a = { a }^{ 2 }$ 
${ a }^{ 2 }=0.8a$

${V} _{ 1 }=\dfrac { 1 }{ 3 } \times { \left( { a } _{1} \right)  }^{ 2 }\times { h } _{ 1 }$ 
${V} _{ 1 }=V$ ....... [Given]

$\Rightarrow \dfrac { 1 }{ 3 } { a }^{ 2 }\times h=\dfrac { 1 }{ 3 } { \left( 0.8 \right)  }^{ 2 }{ a }^{ 2 }\times { h } _{ 1 }$

$\therefore { h } _{ 1 }=1.5625h$ 

$\implies \dfrac { { h } _{ 1 }-h }{ h } =1.5625-1=56.25%$

$\therefore$  'h' need to be increase by $56.25$