Question

Find the approximate change in the value of V of a cube of side $x$ metres caused by increasing the side by $1\% $.

Answer 1

Hint: Firstly, find the new side of the cube by solving the percentage increase in the side and then find new volume of the cube. We will get the change in value of $V$ by comparing the original volume with the new volume.

Formula used: Percentage $ = \dfrac{a}{b} \times 100$


Complete step by step solution:
1. The side of the cube is $x$ metres. New side of the cube after $1\% $ of increase is given as
New side of cube $ = x + x \times 1\% $
$
   = x + x \times \dfrac{1}{{100}} \\
   = x + x(0.01) \\
   = \left( {1.01} \right)x \\
 $
Therefore, new side of the cube after increase is $\left( {1.01} \right)x$
2. Volume of cube when side was $x$meters
$V = {x^3}$
3. Volume of the cube when side becomes $\left( {1.01} \right)x$meters
$V' = {\left\{ {\left( {1.01} \right)x} \right\}^3}$
$V' = {\left( {1.01} \right)^3}{x^3}$
4. Change in volume will be given as difference of step 4 and step 3
Change in V $ = {\left( {1.01} \right)^3}{x^3} - {x^3}$
Change in V $ = \left\{ {{{\left( {1.01} \right)}^3} - 1} \right\}{x^3}$
Change in V $ = \left\{ {1.030301 - 1} \right\}x$
Change in V $ = \left( {0.30301} \right)x$
Hence, approximate change in the value of \[V\]when side $x$ meters change by $1\% $ is $\left( {0.30301} \right)x$


Note: Care has to be taken to see the unit in which a given quantity is expressed and while solving care has to be taken to see that the final answer is expressed in the appropriate unit asked