The length of a side of a cube is 10cm; if an error of 0.05cm is made in measuring the side, then what is the value of percentage error in calculating its volume?

The correct option is 1.5.

Let the length of a side of the cube be represented by $x$.
According to the problem, the measured length of the side is:
$x=10\text{ cm}$

The error made in measuring the side, denoted as $\Delta x$ (or $dx$), is:
$\Delta x=0.05\text{ cm}$

The volume $V$ of a cube with side length $x$ is given by the formula:
$V=x^3$

To find the error in calculating the volume, we differentiate $V$ with respect to $x$:
$\frac{dV}{dx}=3x^2$

This gives the approximate error in volume, $dV$, as:
$dV=3x^{2}dx$

The relative error in the volume is given by dividing the error in volume by the total volume:
$\frac{dV}{V}=\frac{3x^{2}dx}{x^3}=3\frac{dx}{x}$

To find the percentage error in the volume, we multiply the relative error by 100:
$\text{Percentage Error in Volume}=\left(\frac{dV}{V}\times 100\right)\%=3\times \left(\frac{dx}{x}\times 100\right)\%$

Now, we substitute the given values ($x=10$ and $dx=0.05$) into the equation:
$\text{Percentage Error in Volume}=3\times \left(\frac{0.05}{10}\times 100\right)\%$

Simplifying the expression inside the parentheses:
$\frac{0.05}{10}\times 100=0.05\times 10=0.5$

Substituting this back into the percentage error formula:
$\text{Percentage Error in Volume}=3\times 0.5\%=1.5\%$

Therefore, the percentage error in calculating the volume of the cube is 1.5.