Question

(a) Three spheres of equal size are composed of aluminum (density \(\left.=2.70 \mathrm{~g} / \mathrm{cm}^{3}\right),\) silver \(\left(\right.\) density \(\left.=10.49 \mathrm{~g} / \mathrm{cm}^{3}\right)\) and nickel (density \(\left.=8.90 \mathrm{~g} / \mathrm{cm}^{3}\right) .\) List the spheres from lightest to heaviest. (b) Three cubes of equal mass are composed of gold (density \(=19.32 \mathrm{~g} / \mathrm{cm}^{3}\) ), platinum (density \(\left.=21.45 \mathrm{~g} / \mathrm{cm}^{3}\right),\) and lead \(\left(\right.\) density \(\left.=11.35 \mathrm{~g} / \mathrm{cm}^{3}\right) .\) List the cubes from smallest to largest. [Section 1.5\(]\)

Short answer

The short answer for the spheres lightest to heaviest is: Aluminum, Nickel, Silver. The short answer for the cubes smallest to largest is: Platinum, Gold, Lead.

Step-by-step solution

Part (a): Compare the densities of three spheres

First, list out the given densities of each sphere: Aluminum sphere: \(2.70 g/cm^3\) Silver sphere: \(10.49 g/cm^3\) Nickel sphere: \(8.90 g/cm^3\) As these spheres have the same volume, compare their densities in ascending order: 1. Aluminum sphere: \(2.70 g/cm^3\) 2. Nickel sphere: \(8.90 g/cm^3\) 3. Silver sphere: \(10.49 g/cm^3\) So, the spheres are lightest to heaviest: Aluminum, Nickel, Silver.

Part (b): Calculate the volumes of three cubes

Since the cubes are of equal mass, we will calculate the volumes of the cubes and compare those. We will be using the rearranged formula: Volume = Mass/Density Using the given densities: Gold cube: \(19.32 g/cm^3\) Platinum cube: \(21.45 g/cm^3\) Lead cube: \(11.35 g/cm^3\) Since the mass of the cubes is the same for all three cubes, let's call this mass M. Now, we can write the volumes of the cubes as: Gold cube volume: \(V_{gold} = \frac{M}{19.32}\) Platinum cube volume: \(V_{platinum} = \frac{M}{21.45}\) Lead cube volume: \(V_{lead} = \frac{M}{11.35}\) Comparing these volumes in ascending order, we obtain: 1. Platinum cube: \(V_{platinum} = \frac{M}{21.45}\) 2. Gold cube: \(V_{gold} = \frac{M}{19.32}\) 3. Lead cube: \(V_{lead} = \frac{M}{11.35}\) So, the cubes are smallest to largest: Platinum, Gold, Lead.

Key concepts

Volume Calculations

Volume calculations help us find the space that an object occupies. In the exercise, we explore three types of volumes: spheres and cubes. The volume of a sphere or a cube can be crucial in determining which object is lightest or heaviest, smallest or largest, depending on the context.
For spheres, although they have different densities, their volumes are equal. Hence, understanding that density helps in changing the perspective from volume to relative mass. In contrast, we approach cubes by considering their equal mass.
To calculate the volume of the cubes, we rearrange the formula for density: \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \). This rearrangement allows us to see how density directly affects the volume given the same mass. The formula shows that smaller densities result in larger volumes, as seen with the lead cube.

Material Properties

Material properties dictate how different materials behave under certain conditions. In this exercise, the key property we focus on is density. Density measures how heavy an object is for its size, that is, mass per unit volume.
Each material in our problem - aluminum, silver, nickel, gold, platinum, and lead - exhibits unique densities. These densities help determine how heavy or how large an object will be when the shape or mass, respectively, is kept constant. This understanding is essential for working out real-world problems, such as sorting materials based on their weight or size.

• Aluminum is known for its lightness due to its low density.
• Silver and nickel are denser, making them heavier despite having the same volume as the aluminum sphere.
• Comparatively, gold, platinum, and lead densities influence the size of cubes when the mass is constant — with lead being the least dense and thus the largest cube.

Mathematical Reasoning

Mathematical reasoning involves using logic and mathematical concepts to solve problems. In the case presented, we use it to compare the density and volume of materials. The given problem challenges us to think about how densities can affect mass (in spheres) and how they can alter volume (in cubes).
For the spheres, mathematical reasoning allows us to arrange materials from lightest to heaviest based on density, since all spheres have the same volume. Less dense materials weigh less, providing a logical tool for comparison.
In the scenario with cubes, reasoning helps us compute and compare volumes. Even though the same mass is kept constant, the densities dictate the spatial aspect or the volume outcome. By reasoning, we deduce that higher density materials occupy less space, again using logic and proportional relationships highlighted in the formula \( \frac{\text{Mass}}{\text{Density}} \).
This methodical approach ties practical observation with mathematical principles, showcasing how mathematics helps solve everyday problems effectively.