Question

You are working on a project where you need the volume of a box. You take the length, height, and width measurements and then multiply the values together to find the volume. You report the volume of the box as $0.310{\mathrm{m}}^3$. If two of your measurements were $0.7120\mathrm{m}$ and $0.52458\mathrm{m},$ what was the other measurement?

Short answer

The height of the box is approximately $0.87354m$.

Step-by-step solution

Write down the given information

We are given the volume of the box (V) and two dimensions, the length (L) and the width (W). Our task is to find the height (H). V = 0.310 m³ L = 0.7120 m W = 0.52458 m

Use the formula for the volume of a box

We can use the formula for the volume of a box, which is V = L × W × H. In this case, we have V = 0.310 m³, L = 0.7120 m, and W = 0.52458 m. 0.310 m³ = (0.7120 m) × (0.52458 m) × H

Solve for H

Divide both sides of the equation by the product of L and W (0.7120 m × 0.52458 m) to isolate H: H = $\frac{0.310m^3}{(0.7120m)\times (0.52458m)}$

Calculate the value of H

Now, perform the calculation: H = $\frac{0.310m^3}{(0.7120m)\times (0.52458m)}$ = 0.87354 m

Write down the answer

We have found the height of the box: H = 0.87354 m

Key concepts

Dimensional Analysis

Dimensional analysis is a critical method in both mathematics and science, used to convert one kind of unit of measure into another and to check the validity of equations. In our exercise, dimensional analysis helps us ensure that the units of length (meters) combine appropriately to result in a volume measured in cubic meters $m^3$.

When calculating the height of the box using the formula for volume $V=L\times W\times H$, where $V$ is volume, $L$ is length, $W$ is width, and $H$ is height, dimensional analysis is implicitly used. For instance, meters times meters gives us square meters $m^2$, and multiplying by meters again gives us cubic meters $m^3$, maintaining the consistency of units throughout the calculation.

Geometry in Chemistry

While geometry is usually associated with mathematics, it is also a foundational element in the field of chemistry, especially in understanding the shapes and structures of molecules. However, when it comes to volume calculations outside the context of molecules, like our box, the geometric principles are the same.

In our example, geometry serves us by providing formulas for calculating volume. The box, a three-dimensional object, has its volume calculated as the product of its length, width, and height $V=L\times W\times H$. It is the straightforward application of geometric principles that allows us to not only calculate such volumes but also to predict and manipulate the shape and space that objects occupy.

Unit Conversion

Unit conversion is essential when working with different measurement systems or when specific units of measure are required. It's a fundamental skill in both everyday tasks and scientific endeavors. In the context of the exercise provided, unit conversion might not be directly employed as the given dimensions are already in meters, which is coherent with the cubic meters unit used for volume.

However, it is crucial to understand how to convert units because the original measurements might not always be in the desired unit. For example, if we had measurements in centimeters, we would need to convert them to meters before using the volume formula by dividing each length by 100 (since 100 centimeters equals 1 meter). This step ensures that when we calculate the volume, the result is expressed in cubic meters $m^3$, adhering to the International System of Units (SI).