Math, asked by snehasekam11, 3 days ago

If the length of the side of a cube is tripled, show that its volume increases 27 times. If
the length of the side of a cube is halved, what is the change in its volume?

please answer it ​

Answers

Answered by Anonymous
37

Given :

Find the change in volume in following cases :-

1.) If Side is tripled .

2.) If Side is halved .

 \\ \rule{200pt}{3pt}

To Find :

  • Change in volume = ?

 \\ \rule{200pt}{3pt}

Solution :

 \dag \; {\underline{\pmb{\frak{ Formula \; Used \; :- }}}}

  •  {\underline{\boxed{\red{\sf{ Volume \; of \; Cube = {Side}^{3} }}}}}

 \\ \qquad{\rule{150pt}{1pt}}

 \dag \; {\underline{\pmb{\frak{ First \; Case \; :- }}}}

Let us Assume :

 \longrightarrow We will assume that the original side is y units . In this case the volume will be 3y . If we will triple the side we will get the side as 3y . Now, By using the formula we will Calculate the Change in volume .

 \\

  • ➟ Original Side = y units
  • ➟ Tripled Side = y × y × y units
  • ➟ Volume = ?

 \\

Calculating the Volume :

 \begin{gathered} \dashrightarrow \; \; \sf { Volume = {Side}^{3} } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \; \; \sf { Volume = {y \times y \times y}^{3} } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \; \; \sf { Volume = {3y}^{3} } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \; \; \sf { Volume = 3y \times 3y \times 3y } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \; \; \sf { Volume = 9y \times 3y } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \; \; {\qquad{\blue{\sf { Volume = 27y }}}} \\ \end{gathered}

 \\ \qquad{\rule{150pt}{1pt}}

 \dag \; {\underline{\pmb{\frak{ Therefore \; :- }}}}

❛❛ If the side of the Cube is tripled its volume will be increased 27 times . ❜❜

 \\ {\underline{\rule{200pt}{5pt}}}

 \dag \; {\underline{\pmb{\frak{ Formula \; Used \; :- }}}}

  •  {\underline{\boxed{\red{\sf{ Volume \; of \; Cube = {Side}^{3} }}}}}

 \\ \qquad{\rule{150pt}{1pt}}

 \dag \; {\underline{\pmb{\frak{ First \; Case \; :- }}}}

 \\

Let Us Assume :

 \longrightarrow We will assume that the original side is y units . In this case the volume will be 3y . If we will half the side we will get the side as  {\bold{\sf{ \dfrac{1}{2}y }}} . Now, By using the formula we will Calculate the Change in volume .

 \\

  • ➟ Original Side = y units

  • ➟ Halved Side =  {\sf{ \dfrac{1}{2}y }}

  • ➟ Volume = ?

 \\

Calculating the Volume :

 \begin{gathered} \dashrightarrow \; \; \sf { Volume = {Side}^{3} } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \; \; \sf { Volume = { \bigg\lgroup \dfrac{1}{2}y \bigg\rgroup }^{3} } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \; \; \sf { Volume = \bigg\lgroup \dfrac{1}{2}y \times \dfrac{1}{2}y \times \dfrac{1}{2}y \bigg\rgroup } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \; \; \sf { Volume = \bigg\lgroup \dfrac{1}{4}y \times \dfrac{1}{2}y \bigg\rgroup } \\ \end{gathered}

 \begin{gathered} \dashrightarrow \; \; {\qquad{\orange{\sf { Volume =  \dfrac{1}{8}y }}}} \\ \end{gathered}

 \\ \qquad{\rule{150pt}{1pt}}

 \dag \; {\underline{\pmb{\frak{ Therefore \; :- }}}}

❛❛ If the side of the Cube is halved its volume will decrease to  {\sf{ \dfrac{1}{8} }} of it's original volume . ❜❜

 \\ {\underline{\rule{200pt}{5pt}}}

Answered by prathameshswami17
0

Step-by-step explanation:

Given :

Find the change in volume in following cases :-

1.) If Side is tripled .

2.) If Side is halved .

\begin{gathered} \\ \rule{200pt}{3pt} \end{gathered}

To Find :

Change in volume = ?

\begin{gathered} \\ \rule{200pt}{3pt} \end{gathered}

Solution :

\dag \; {\underline{\pmb{\frak{ Formula \; Used \; :- }}}}†

FormulaUsed:−

FormulaUsed:−

{\underline{\boxed{\red{\sf{ Volume \; of \; Cube = {Side}^{3} }}}}}

VolumeofCube=Side

3

\begin{gathered} \\ \qquad{\rule{150pt}{1pt}} \end{gathered}

\dag \; {\underline{\pmb{\frak{ First \; Case \; :- }}}}†

FirstCase:−

FirstCase:−

Let us Assume :

\longrightarrow⟶ We will assume that the original side is y units . In this case the volume will be 3y . If we will triple the side we will get the side as 3y . Now, By using the formula we will Calculate the Change in volume .

\begin{gathered} \\ \end{gathered}

➟ Original Side = y units

➟ Tripled Side = y × y × y units

➟ Volume = ?

\begin{gathered} \\ \end{gathered}

Calculating the Volume :

\begin{gathered} \begin{gathered} \dashrightarrow \; \; \sf { Volume = {Side}^{3} } \\ \end{gathered} \end{gathered}

⇢Volume=Side

3

\begin{gathered} \begin{gathered} \dashrightarrow \; \; \sf { Volume = {y \times y \times y}^{3} } \\ \end{gathered} \end{gathered}

⇢Volume=y×y×y

3

\begin{gathered} \begin{gathered} \dashrightarrow \; \; \sf { Volume = {3y}^{3} } \\ \end{gathered} \end{gathered}

⇢Volume=3y

3

\begin{gathered} \begin{gathered} \dashrightarrow \; \; \sf { Volume = 3y \times 3y \times 3y } \\ \end{gathered} \end{gathered}

⇢Volume=3y×3y×3y

\begin{gathered} \begin{gathered} \dashrightarrow \; \; \sf { Volume = 9y \times 3y } \\ \end{gathered} \end{gathered}

⇢Volume=9y×3y

\begin{gathered} \begin{gathered} \dashrightarrow \; \; {\qquad{\blue{\sf { Volume = 27y }}}} \\ \end{gathered} \end{gathered}

⇢Volume=27y

\begin{gathered} \\ \qquad{\rule{150pt}{1pt}} \end{gathered}

\dag \; {\underline{\pmb{\frak{ Therefore \; :- }}}}†

Therefore:−

Therefore:−

❛❛ If the side of the Cube is tripled its volume will be increased 27 times . ❜❜

\begin{gathered} \\ {\underline{\rule{200pt}{5pt}}} \end{gathered}

\dag \; {\underline{\pmb{\frak{ Formula \; Used \; :- }}}}†

FormulaUsed:−

FormulaUsed:−

{\underline{\boxed{\red{\sf{ Volume \; of \; Cube = {Side}^{3} }}}}}

VolumeofCube=Side

3

\begin{gathered} \\ \qquad{\rule{150pt}{1pt}} \end{gathered}

\dag \; {\underline{\pmb{\frak{ First \; Case \; :- }}}}†

FirstCase:−

FirstCase:−

\begin{gathered} \\ \end{gathered}

Let Us Assume :

\longrightarrow⟶ We will assume that the original side is y units . In this case the volume will be 3y . If we will half the side we will get the side as {\bold{\sf{ \dfrac{1}{2}y }}}

2

1

y . Now, By using the formula we will Calculate the Change in volume .

\begin{gathered} \\ \end{gathered}

➟ Original Side = y units

➟ Halved Side = {\sf{ \dfrac{1}{2}y }}

2

1

y

➟ Volume = ?

\begin{gathered} \\ \end{gathered}

Calculating the Volume :

\begin{gathered} \begin{gathered} \dashrightarrow \; \; \sf { Volume = {Side}^{3} } \\ \end{gathered} \end{gathered}

⇢Volume=Side

3

\begin{gathered} \begin{gathered} \dashrightarrow \; \; \sf { Volume = { \bigg\lgroup \dfrac{1}{2}y \bigg\rgroup }^{3} } \\ \end{gathered} \end{gathered}

⇢Volume=

2

1

y

3

\begin{gathered} \begin{gathered} \dashrightarrow \; \; \sf { Volume = \bigg\lgroup \dfrac{1}{2}y \times \dfrac{1}{2}y \times \dfrac{1}{2}y \bigg\rgroup } \\ \end{gathered} \end{gathered}

⇢Volume=

2

1

2

1

2

1

y

\begin{gathered} \begin{gathered} \dashrightarrow \; \; \sf { Volume = \bigg\lgroup \dfrac{1}{4}y \times \dfrac{1}{2}y \bigg\rgroup } \\ \end{gathered} \end{gathered}

⇢Volume=

4

1

2

1

y

\begin{gathered} \begin{gathered} \dashrightarrow \; \; {\qquad{\orange{\sf { Volume = \dfrac{1}{8}y }}}} \\ \end{gathered} \end{gathered}

⇢Volume=

8

1

y

\begin{gathered} \\ \qquad{\rule{150pt}{1pt}} \end{gathered}

\dag \; {\underline{\pmb{\frak{ Therefore \; :- }}}}†

Therefore:−

Therefore:−

❛❛ If the side of the Cube is halved its volume will decrease to {\sf{ \dfrac{1}{8} }}

8

1

of it's original volume . ❜❜

\begin{gathered} \\ {\underline{\rule{200pt}{5pt}}} \end{gathered}

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