Question

The inside dimension of a box that is cubic is \(24.8 \mathrm{~cm}\) on each edge with an uncertainty of \(0.2 \mathrm{~cm}\). What is the volume of the box? What do you estimate to be the uncertainty in the calculated volume?

Short answer

The volume of the cubic box is approximately \(15,287.744 \mathrm{~cm^3}\), with an estimated uncertainty of \(369.248 \mathrm{~cm^3}\).

Step-by-step solution

Find the volume of the cube

Use the volume formula V = a³, where "a" is the length of the edge given (24.8 cm). Substitute the given value in the formula and find the volume: V = (24.8 cm)³ V = 15,287.744 cm³ The volume of the box is 15,287.744 cubic centimeters.

Calculate the partial derivative of the volume formula

We will differentiate the volume formula with respect to the length "a": dV/da = 3a² Now substitute the given edge length (24.8 cm) in the derivative formula: dV/da = 3(24.8 cm)² dV/da = 1,846.24 cm²

Find the uncertainty in the volume

To find the uncertainty in the calculated volume, we will use the propagation of uncertainty formula: ΔV = |dV/da| * Δa Here, Δa is the uncertainty in the given edge length (0.2 cm). Substitute the values in the formula and find the uncertainty in the volume: ΔV = 1,846.24 cm² * 0.2 cm ΔV = 369.248 cm³ The uncertainty in the calculated volume is 369.248 cubic centimeters. To summarize, the volume of the box is approximately 15,287.744 cm³, with an estimated uncertainty of 369.248 cm³.

Key concepts

Understanding Measurement Uncertainty

Measurement uncertainty represents the doubt that exists about the result of any measurement. No measurement can be absolutely certain, and there's always a margin of error. In this exercise, when determining the length of the cube's edge as 24.8 cm, there is an uncertainty of 0.2 cm. This tells us that the actual length could be slightly longer or shorter than the measured value. Considering measurement uncertainty is crucial because:

• It allows for the estimation of how much the measured value could deviate from the true value.
• It gives a range within which the true measurement likely lies.
• It helps in understanding the reliability and accuracy of the measurement process.

Understanding and calculating uncertainty helps ensure realistic and meaningful experimental results.

The Role of Partial Derivatives

Partial derivatives help understand how a function changes as its variables change. In the context of volume calculation, we use partial derivatives to see how a small change in the edge length affects the volume. Here, the volume of a cube is given by the formula \(V = a^3\), which depends solely on the edge length \(a\).To get the partial derivative \(\frac{dV}{da}\), we differentiate \(V = a^3\) with respect to \(a\), which yields \(\frac{dV}{da} = 3a^2\).Partial derivatives show us:

• The rate of change of volume concerning the edge length.
• How sensitive the volume is to changes in the edge length.
• They are used in the propagation of uncertainty to determine how errors in measurement propagate through calculations.

This tool is essential in calculus and beyond, offering insights into the relationships between different variables.

Grasping Propagation of Uncertainty

Propagation of uncertainty quantifies how uncertainty in measurements affects the results obtained through calculations. In other words, it allows us to calculate the uncertainty of a derived quantity, like volume, based on the uncertainty in the measured variables, like the edge length.The propagation of uncertainty formula used here is \(\Delta V = |\frac{dV}{da}| \times \Delta a\), where \(\Delta a\) is the uncertainty in \(a\). For the box's volume, \(\Delta a = 0.2\) cm, and \(\frac{dV}{da} = 1,846.24\) cm². Substituting these values, \(\Delta V = 1,846.24 \times 0.2 = 369.248\) cm³.This concept helps us:

• Establish the uncertainty range for computed measurements.
• Ensure precise and accurate reporting of calculated quantities.
• Account for all possible measurement errors in scientific work.

A deep understanding of propagation of uncertainty is essential in scientific and engineering fields.

The Importance of the Cubic Centimeter

A cubic centimeter is a unit of volume widely used in the metric system. It is especially useful for measuring smaller volumes. When we calculate the volume of the box, our result is 15,287.744 cubic centimeters. This unit is significant because:

• It is a convenient measure for small and medium-sized objects.
• It is widely recognized and used internationally.
• It provides a clear connection between linear measures (cm) and volumes (cm³).

Understanding the use of cubic centimeters helps in grasping the scale and applicability of volume measurements across various scientific and practical fields.