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
Given :
Find the change in volume in following cases :-
1.) If Side is tripled .
2.) If Side is halved .
To Find :
- Change in volume = ?
Solution :
Let us Assume :
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 :
❛❛ If the side of the Cube is tripled its volume will be increased 27 times . ❜❜
Let Us Assume :
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 . Now, By using the formula we will Calculate the Change in volume .
- ➟ Original Side = y units
- ➟ Halved Side =
- ➟ Volume = ?
Calculating the Volume :
❛❛ If the side of the Cube is halved its volume will decrease to of it's original volume . ❜❜
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
y×
2
1
y×
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
y×
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}