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 approximate error in calculating its volume?

The correct option is 15 cu cm.

Let us understand how to find the approximate error in calculating the volume of a cube using differentials.

Step 1: Define the variables and the given values
Let $x$ be the length of the side of the cube.
Let $V$ be the volume of the cube.
We are given:
The length of the side of the cube, $x=10\text{ cm}$.
The error in measuring the side, $dx=\Delta x=0.05\text{ cm}$.

Step 2: Formula for the volume of a cube
The volume $V$ of a cube of side $x$ is given by the formula:

$V=x^3$

Step 3: Differentiate the volume with respect to the side length
To find the rate of change of volume with respect to $x$, we differentiate both sides with respect to $x$:

$\frac{dV}{dx}=\frac{d}{dx}\left(x^3\right)=3x^2$

Step 4: Calculate the approximate error in volume
The approximate error in volume, denoted by $dV$ (or $\Delta V$), is given by:

$dV=\left(\frac{dV}{dx}\right)·dx$

Substituting the derivative from Step 3:

$dV=3x^2·dx$

Step 5: Substitute the given numerical values
Now, we substitute $x=10$ and $dx=0.05$ into the equation:

$dV=3·{\left(10\right)}^2·\left(0.05\right)$

$dV=3·100·0.05$

$dV=300·0.05$

$dV=15\text{ cu cm}$

Thus, the approximate error in calculating the volume of the cube is 15 cu cm.