Question

The volume of a square pyramid is \((1 / 3) \mathrm{B} h\) where \(\mathrm{B}\) is the area of the square base and \(h\) is the height. One student's measurements of a pyramid are \(\mathrm{B}=1.33 \mathrm{~m}^{2}, h=2.79 \mathrm{~m} .\) Another student's measurements of the same pyramid are \(\mathrm{B}=1.31 \mathrm{~m}^{2}, h=2.81 \mathrm{~m} .\) What is the difference in the volumes of the pyramid calculated by the two students?

Short answer

Answer: The difference in the volumes of the pyramid calculated by the two students is 0.0096 m³.

Step-by-step solution

Find the volume using the first student's measurements

Using the formula \(V = \frac{1}{3}Bh\), plug in the first student's measurements: \(V_1 = \frac{1}{3}(1.33)(2.79)\) \(V_1 = \frac{1}{3}(3.7097)\) \(V_1 = 1.2366 \mathrm{~m}^3\)

Find the volume using the second student's measurements

Similarly, use the formula with the second student's measurements: \(V_2 = \frac{1}{3}(1.31)(2.81)\) \(V_2 = \frac{1}{3}(3.6811)\) \(V_2 = 1.2270 \mathrm{~m}^3\)

Calculate the difference in volumes

Subtract the smaller volume from the larger volume to find the difference: Difference \(= V_1 - V_2 = 1.2366 - 1.2270 = 0.0096 \mathrm{~m}^3\) So, the difference in the volumes of the pyramid calculated by the two students is \(0.0096 \mathrm{~m}^3\).

Key concepts

Mathematics in Chemistry

Mathematics plays an essential role in chemistry, oftentimes because it provides tools to quantify and predict chemical phenomena. For instance, volume calculation is paramount when adjusting reagents in a reaction or when measuring substances in a lab. Chemistry deals with both pure substances and mixtures, where the understanding of volume, alongside concentration and density, is vital for successful experimentation and analysis.

Mathematical expressions and equations, like the volume of a square pyramid \( V = \frac{1}{3}Bh \), allow chemists to calculate the space a substance occupies or to design containers for storage and reaction purposes. These calculations can further apply to materials science, pharmacology, and engineering, showcasing the interdisciplinary nature of mathematics in scientific inquiry. In practical chemistry scenarios, it's often necessary to approximate real-life irregular shapes to geometrical ones, like approximating crystals or molecules to pyramidal structures, to facilitate volume calculations.

Volume Calculation

The volume of a three-dimensional object is a measure of the space it occupies, which is vital in various fields such as architecture, engineering, and science. The volume of regular geometric shapes like cubes, cylinders, and pyramids can be found using specific formulas. For a square pyramid, which has a square base and four triangular faces converging to a point, the formula \( V = \frac{1}{3}Bh \) is used, where \(B\) is the area of the base and \(h\) is the pyramid's height.

Understanding how to calculate volume is critical not only for solving textbook problems but also for real-world applications. For example, calculating the volume of a storage tank, understanding how much space is left in a container, or utilizing space efficiently all require volume calculations. The step-by-step calculation method provided in the solution illustrates how to substitute given values into the formula to find a precise volume.

Measurement Uncertainty

Measurement uncertainty refers to the doubt that exists regarding the value of a measurement. Regardless of precision and accuracy efforts in science, every measurement has a degree of uncertainty. Factors such as the limitations of the measuring instrument, the skill of the person conducting the measurement, and external conditions can contribute to measurement uncertainty.

This uncertainty is relevant in the exercise, as two students measuring the same pyramid have produced slightly different base areas and heights, leading to different volume calculations. While the measurements are close, the variances are normal and expected due to measurement uncertainty. It highlights the importance of being aware that measurements are not exact and that calculated results based on these measurements will also carry this margin of error. In practical applications, understanding measurement uncertainty helps in making allowances for potential discrepancies and is crucial for quality control and data reliability.