Question

You win the lottery and decide to impress your friends by exhibiting a million-dollar cube of gold. At the time, gold is selling for $1282 per troy ounce, and 1.0000 troy ounce equals 31.1035 g. How tall would your million- dollar cube be?

Short answer

The cube of gold would be approximately 10.78 cm (or 4.24 inches) tall.

Step-by-step solution

Calculate Mass of Gold in Troy Ounces

First, determine how many troy ounces of gold you can buy with one million dollars. Since gold costs $1282 per troy ounce, you can find the number of troy ounces by dividing the total money by the price per ounce: \[ \text{Troy ounces} = \frac{1,000,000}{1282} \approx 780.03 \text{ troy ounces} \]

Convert Troy Ounces to Grams

Each troy ounce is equivalent to 31.1035 grams. Convert the total gold weight from troy ounces to grams using this conversion factor:\[ \text{Grams of gold} = 780.03 \times 31.1035 \approx 24,272.50 \text{ grams} \]

Calculate Volume of Gold in Cubic Centimeters

Find the volume of the gold by dividing its mass in grams by the density of gold (19.32 g/cm³):\[ \text{Volume} = \frac{24,272.50 \text{ grams}}{19.32 \text{ g/cm}^3} \approx 1,256.82 \text{ cm}^3 \]

Determine Side Length of the Cube

The volume of a cube is given by the cube of its side length. Therefore, solve for the side length using:\[ s^3 = 1,256.82 \]Thus, the side length is:\[ s = \sqrt[3]{1,256.82} \approx 10.78 \text{ cm} \]

Convert Side Length to Inches

Convert the side length from centimeters to inches by dividing by 2.54 (since there are 2.54 cm in an inch):\[ s \approx \frac{10.78}{2.54} \approx 4.24 \text{ inches} \]

Key concepts

Troy Ounce

A troy ounce is a unit of measure often used in the precious metals market, including gold, silver, and platinum. It's crucial to understand that a troy ounce is not the same as a regular ounce. While a troy ounce equals approximately 31.1035 grams, a standard ounce (often referred to as an avoirdupois ounce) is only about 28.35 grams.
This distinction is important since pricing and weight calculations in the context of precious metals rely on the troy ounce. Knowing the weight difference helps avoid confusion and ensures precise conversions and calculations in financial contexts.

Density

Density is a key concept in physics and chemistry that connects mass and volume. It is defined as the mass of an object divided by its volume, usually expressed in grams per cubic centimeter (g/cm³) for solids and liquids. In this scenario, the density of gold is provided as 19.32 g/cm³.
Understanding density allows us to determine how much space a certain mass of material will occupy. It can also be helpful in identifying substances based on their mass-to-volume ratio, as different materials have distinct densities depending on their atomic structure and composition.

• High density means a substance is heavy for its size.
• Low density indicates a substance is light for its size.

Volume of a Cube

The volume of a cube is calculated by taking the cube of its side length. This is expressed mathematically as: \[ V = s^3 \] where \( V \) represents volume and \( s \) denotes the side length of the cube.
Solving this equation allows us to find any missing dimension, provided we have the other two. In this exercise, knowing the volume of gold in cubic centimeters enables us to solve for the side length of the billion-dollar gold cube by applying the cube root to the volume. This emphasizes why spatial reasoning is essential in geometry and real-world applications.

Unit Conversion

In many scientific problems, converting units accurately is indispensable for obtaining correct results. Units conversion involves transforming one unit of measurement into another without changing the original value or quantity of the measurement.
For example, in the exercise: - Gold's weight in troy ounces was first converted to grams. - Then, dimensions were converted from centimeters to inches (used for the final side measurement of the cube).
These conversions ensure that we are working with units that are appropriate and consistent within calculations, bridging different systems of measurement effectively. Converting units helps in understanding and comparing different types of measurements without altering the significance of the data involved.