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 relative error in calculating its volume?

The correct option is 0.015.

To find the relative error in calculating the volume of the cube, we can use differentials. Let us break down the solution step-by-step.

Step 1: Understand the given values
Let the side length of the cube be represented by $x$.
Given length of the side of the cube:
$$x=10\text{ cm}$$
The error made in measuring the side, which we can denote as $dx$ (or $\Delta x$):
$$dx=0.05\text{ cm}$$

Step 2: Relate volume to the side length
The formula for the volume $V$ of a cube with side length $x$ is:
$$V=x^3$$

Step 3: Differentiate to find the error in volume
Taking the derivative of volume $V$ with respect to $x$, we get:
$$\frac{dV}{dx}=3x^2$$
Therefore, the differential error in volume, $dV$, is:
$$dV=3x^{2}dx$$

Step 4: Calculate the relative error in volume
The relative error in volume is defined as the ratio of the error in volume ($dV$) to the original volume ($V$):
$$\text{Relative Error}=\frac{dV}{V}$$
Substitute the expressions for $dV$ and $V$ into the equation:
$$\frac{dV}{V}=\frac{3x^{2}dx}{x^3}$$
Simplify the expression by dividing the terms:
$$\frac{dV}{V}=3\frac{dx}{x}$$

Step 5: Substitute the given numerical values
Now, substitute $x=10$ and $dx=0.05$ into the simplified relative error formula:
$$\frac{dV}{V}=3\times \frac{0.05}{10}$$
Simplify the division inside the expression:
$$\frac{dV}{V}=3\times 0.005$$
Calculate the final product:
$$\frac{dV}{V}=0.015$$

Thus, the relative error in calculating the volume of the cube is 0.015.