Problem 29 - Chemistry: Principles and Reactions

Question

The volume of a sphere is \(4 \pi r^{3} / 3\), where \(r\) is the radius. One student measured the radius to be \(4.30 \mathrm{~cm} .\) Another measured the radius to be \(4.33 \mathrm{~cm}\). What is the difference in volume between the two measurements?

Step-by-Step Solution

Answer

Answer: The difference in volume between the two spheres is approximately 6.3 cubic centimeters.

Step 1: Calculate the volume of the first sphere

To find the volume of the first sphere, we use the given formula and plug in the first radius measurement, \(4.3\mathrm{~cm}\). The formula we apply is: $$ V_1 = \frac{4}{3}\pi (4.3)^3 $$

Step 2: Calculate the volume of the second sphere

Similarly, to find the volume of the second sphere, we use the same formula and plug in the second radius measurement, \(4.33\mathrm{~cm}\). The formula we apply is: $$ V_2 = \frac{4}{3}\pi (4.33)^3 $$

Step 3: Calculate the difference in volume between the two spheres

Once we have both volumes, we can calculate the difference between them by finding the absolute value of the difference. The formula we use is: $$ \Delta V = |V_2 - V_1| $$

Step 4: Calculate the volumes and difference

Now we can calculate the volumes of both spheres along with their difference: $$ V_1 = \frac{4}{3}\pi (4.3)^3 \approx 333.04\;\mathrm{cm}^3 $$ $$ V_2 = \frac{4}{3}\pi (4.33)^3 \approx 339.34\;\mathrm{cm}^3 $$ $$ \Delta V = |339.34 - 333.04| \approx 6.3\;\mathrm{cm}^3 $$ The difference in volume between the two measurements is approximately \(6.3\;\mathrm{cm}^3\).

Key Concepts

Volume Formula for Sphere

Calculating the volume of a sphere is a fundamental concept in geometry, which can be especially helpful in various practical scenarios like determining the amount of liquid that a spherical container can hold. The general formula for the volume of a sphere is \( V = \frac{4}{3}\pi r^{3} \) where \( r \) is the radius of the sphere. The radius is simply the distance from the center of the sphere to any point on its surface.

As we delve into this formula, it is crucial to recognize the role of the cube function (raising to the power of three) in this equation. This function indicates that the volume of a sphere increases rapidly with an increase in the radius. Consequently, even a small change in the radius can significantly alter the volume, which is evident in the provided exercise comparing two slightly different radius measurements for two spheres.

Measurement Precision

Measurement precision is crucial when working with geometrical formulas. When measuring a physical object, there's always a degree of uncertainty, and even a minuscule error can lead to different outcomes. In the context of the exercise, two students measured the radius of a sphere and obtained slightly different values: \(4.30 \text{cm}\) and \(4.33 \text{cm}\).

This underscores the importance of precise measurements and the proper use of significant figures. It is also vital to use accurate instruments and techniques to minimize errors when measuring objects, as these errors propagate through calculations and impact the final result. An understanding of how precision affects computations is a valuable skill in fields such as engineering, physics, and even everyday activities.

Comparing Volumes

Comparing volumes is a key task in geometry that can help us understand relationships between different objects. It involves using the volume formula for spheres and taking the absolute difference of the results. As seen in the step-by-step solution, the difference in volume between two spheres with radii \(4.30 \text{cm}\) and \(4.33 \text{cm}\) is approximately \(6.3 \text{cm}^3\).

To compare volumes effectively, one must not only be proficient in using the formulas but also understand the concepts of measurement precision. This is because even a slight measurement error can lead to a significant difference in the calculated volumes. Comprehending these differences is essential for tasks such as optimizing packaging, filling materials, and understanding how spheres behave when dimensions change.