Question

Percentage Error The edge of a cube is measured as 10 \(\mathrm{cm}\) with an error of 1\(\%\) . The cube's volume is to be calculated from this measurement. Estimate the percentage error in the volume calculation.

Short answer

The percentage error in the volume calculation is 3%.

Step-by-step solution

Understanding the Concept

For a cube, the volume, \(V\), is given by \(V = a^3\), where \(a\) is the edge length. Differentiating this with respect to \(a\), we get \(\frac{dV}{da} = 3a^2\). The ratio \(\frac{\Delta V}{V}\), i.e., the fractional change in volume, is thus three times the fractional change in \(a\), i.e., \(\frac{\Delta a}{a}\), as the cube's volume changes three times as fast as its edge length for a small change.

Calculating the Percentage Error in Volume

The relative error, or percentage error, is found by multiplying the fractional error by 100%. The edge length is given with an error of 1%. Therefore, the percentage error in the calculation of the volume is three times this, i.e., \(3 \times 1\% = 3\%\).

Key concepts

Relative Error

Relative error is a measure of the precision of a measurement compared to the actual value. It is a crucial concept in many scientific and engineering calculations, as it gives an idea of how much uncertainty there is in a measured value.

When we talk about the 'relative error' in a measurement, we are referring to the absolute error divided by the true value. For instance, if we have a true value of an object's length as 10 cm and a measured value of 10.1 cm, the absolute error is 0.1 cm. Dividing this by the true value (10 cm), we get the relative error as 0.01 or 1%.

This concept extends to any calculations made using the measured value, such as volume. If we know the relative error of a length measurement, we can use it to estimate the relative error in a calculation derived from that length, like determining the volume of a cube.

Differential Calculus

Differential calculus is a branch of mathematics that deals with the rates at which quantities change. It is the foundation for calculating error propagation, especially when the relationship between variables is not straightforward.

In our context, we use differential calculus to understand how small changes in the measurement of an edge of a cube will affect the volume of the cube. The derivative can be thought of as an instantaneous rate of change, or how a function's output changes as its input changes. In simpler terms, if we have a small error in measuring an edge, calculus helps us to determine how magnified this error will be when we calculate the cube's volume.

By differentiating the formula for the volume of a cube, we establish a relationship between the change in volume and the change in edge length, leading us to how these errors translate from lengths to volumes.

Volume of a Cube

The volume of a cube is a fundamental geometric concept that reflects how much space a cube occupies. It is calculated by the formula

\[ V = a^3 \]
Where 'V' represents the volume and 'a' is the length of one edge of the cube.

Understanding this cubic relationship is critical because it tells us that the volume of a cube changes dramatically with even the slightest change in edge length. This is especially important in the context of errors in measurement, as a small measurement error on the edge length translates into a significantly larger error in volume due to this cubic nature.

Fractional Change

Fractional change is a way to express how much a quantity changes in relation to its original value. It is often presented as a ratio or a percentage.

In practical scenarios, like the cube's volume calculation, the fractional change is directly related to measurement errors. Small errors in measuring the edge length of a cube lead to fractional changes in the length itself. Since the volume depends on the cube of the length, these fractional changes are cubed, leading to a more considerable change in volume.

For example, a 1% increase in the edge length of a cube (fractional change of 0.01) results in a 3% increase in volume (fractional change of 0.03). This is why it is vital not only to measure as accurately as possible but also to understand the implications of these measurements on calculated quantities.