This question aims to develop an understanding of the algebraic simplification of the equation for the volume of a block using basic arithmetic operations.
The volume of a block is the product of its length, width, and height. It is mathematically defined by the following formula:
\[ \boldsymbol{ V \ = \ L \times W \times H } \]
Where $ V $ represents the volume of the block, $ L $ represents the length, $ W $ represents the width, and $ H $ represents the height. Now this formula can be directly used to calculate the volume given the length, width and height of the block, however, if we were to evaluate the value of $ h $ given the volume, then we may have to modify it a little bit. This rearrangement process is called the algebraic simplification process, which is further explained in the following solution.
Expert Answer
Given the formula of the volume of the block:
\[ V \ = \ L \times W \times H \]
Dividing both sides by $ W $:
\[ \dfrac{ V }{ W } \ = \ \dfrac{ L \times W \times H }{ W } \]
\[ \Rightarrow \dfrac{ V }{ W } \ = \ L \times H \]
Dividing both sides by $ H $:
\[ \dfrac{ V }{ W \times H } \ = \ \dfrac{ L \times H }{ H } \]
\[ \Rightarrow \dfrac{ V }{ W \times H } \ = \ L \]
Exchanging sides:
\[ L \ = \ \dfrac{ V }{ W \times H } \]
Which is the required expression.
Numerical Result
\[ L \ = \ \dfrac{ V }{ W \times H } \]
Example
Part (a) – The area of a rectangle is given by the following formula:
\[ A \ = \ L \times W \]
Find the value of $ L $.
Dividing the above equation by $ W $:
\[ \dfrac{ A }{ W } \ = \ \dfrac{ L \times W }{ W } \]
\[ \Rightarrow \dfrac{ A }{ W } \ = \ L \]
Exchanging sides:
\[ L \ = \ \dfrac{ A }{ W } \]
Part (b) – The area of a right triangle is given by the following formula:
\[ A \ = \ \dfrac{ 1 }{ 2 } b \times h \]
Find the value of $ h $.
Dividing the above equation by $ b $:
\[ \dfrac{ A }{ b } \ = \ \dfrac{ 1 }{ 2 } \dfrac{ b \times h }{ b } \]
\[ \Rightarrow \dfrac{ A }{ b } \ = \ \dfrac{ 1 }{ 2 } h \]
Multiplying the above equation with $ 2 $:
\[ 2 \times \dfrac{ A }{ b } \ = \ 2 times \dfrac{ 1 }{ 2 } h \]
\[ \Rightarrow 2 \times \dfrac{ A }{ b } \ = \ h \]
Exchanging sides:
\[ h \ = \ 2 \times \dfrac{ A }{ b } \]