Problem 72 - Chemistry The Central Science

Question

A thief plans to steal a gold sphere with a radius of \(28.9 \mathrm{~cm}\) from a museum. If the gold has a density of \(19.3 \mathrm{~g} / \mathrm{cm}^{3},\) what is the mass of the sphere in pounds? [The volume of a sphere is \(\left.V=(4 / 3) \pi r^{3} .\right]\) Is the thief likely to be able to walk off with the gold sphere unassisted?

Step-by-Step Solution

Answer

The mass of the gold sphere is approximately \(4300.2 \text{lb}\). It is highly unlikely that the thief will be able to walk off unassisted with such a heavy object.

Step 1: Calculate the volume of the sphere.

Using the given formula for the volume of a sphere, we will substitute the radius (r) with the given value: \(28.9 \text{cm}\). \[V = \frac{4}{3} \pi r^3\] \[V = \frac{4}{3} \pi (28.9)^3\] Now that we have the formula set up, we will compute the volume.

Step 2: Compute the volume of the sphere

We will input the equation into the calculator and find the value of the volume V. \[V \approx 101207 \text{cm}^3\] The volume of the gold sphere is approximately \(101207 \text{cm}^3\).

Step 3: Calculate the mass of the sphere

We have the density of gold which is \(19.3 \text{g/cm}^3\), and we will multiply it by the volume to find the mass. Mass = (Density of Gold) × (Volume of the Sphere) Mass \(= 19.3 \times 101207\) Now we will compute the mass.

Step 4: Compute the mass of the sphere

Input the equation into the calculator and find the value of the mass. Mass \(\approx 1951299.1 \text{g}\) The mass of the gold sphere is approximately \(1951299.1 \text{g}\).

Step 5: Convert the mass from grams to pounds

We know that 1 pound (lb) is equal to \(453.592 \text{g}\). Therefore, to convert the mass of the sphere from grams to pounds, we will divide the mass in grams by \(453.592\). Mass \(\approx \frac{1951299.1}{453.592}\) Compute the mass in pounds.

Step 6: Compute the mass in pounds

Input the equation into the calculator and find the value of the mass in pounds. Mass \(\approx 4300.2 \text{lb}\) The mass of the gold sphere is approximately \(4300.2 \text{lb}\).

Step 7: Determine if the thief can walk off unassisted with the gold sphere

A gold sphere weighing \(4300.2 \text{lb}\) is very heavy. It is highly unlikely that the thief will be able to walk off unassisted with such a heavy object.

Key Concepts

Volume of a Sphere

Understanding the volume of a sphere is crucial for solving many real-world problems, ranging from determining the amount of liquid a spherical container can hold to identifying the mass of a solid sphere, like the gold sphere in our exercise.

The mathematical formula for the volume of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \] where \( V \) represents the volume and \( r \) stands for the radius of the sphere. The constant \( \pi \) is approximately 3.14159, and it reflects the ratio of the circumference of a circle to its diameter. This formula is derived from the calculus and provides an exact number for the volume based on the sphere's radius.

To visualize it more simply, imagine filling a spherical balloon with water. The amount of water required to completely fill the balloon represents its volume. Interestingly, if we were to compare spheres of different sizes, a sphere with double the radius would not just have double the volume, but eight times as much! This is because the volume of a sphere increases with the cube of the radius.

Mass Conversion

Mass conversion is the process of changing the unit of mass from one system of measurement to another. In the context of our textbook exercise, it involves converting the calculated mass of the gold sphere from grams to pounds. This step is important because different regions and fields use different units, and being adept at converting between them is a valuable skill.

To perform the conversion, we use a conversion factor. Here is the relation between grams and pounds: \[ 1 \text{lb} = 453.592 \text{g} \] This means that one pound is equal to 453.592 grams. By dividing the mass value in grams by this number, we get the mass in pounds. It is essential to be precise and ensure that the conversion factor is used correctly to avoid errors in calculation.

Furthermore, being able to convert mass also allows us to relate to the values better because, for instance, in the United States, consumers are more familiar with pounds as compared to grams, which are commonly used elsewhere.

Unit of Measurement

Units of measurement are standardized quantities used to express physical properties like volume, mass, length, and time. They serve as a crucial language in science, engineering, trade, and daily life, allowing us to communicate and compare quantities effectively.

Standard units such as meters for length, kilograms for mass, or seconds for time, are defined by international systems like the International System of Units (SI). However, other measurement systems, such as the Imperial system, are also in use in specific regions, using pounds for mass and feet for length.

Using the correct unit of measurement and being consistent is vital for accuracy and clarity. Similarly, converting units, as needed in our exercise, requires a strong grasp of these units and their conversion factors to ensure the correct outcomes. For instance, the thief in the gold sphere problem would likely think of the weight of their loot in pounds rather than grams, given it's the conventional unit in many everyday contexts.