Gold Brick.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?

검증된 단계별 안내

First, determine the total mass of gold you can buy with one million dollars. Use the price of gold per troy ounce to find the total number of troy ounces you can purchase: $ \text{Total troy ounces} = \frac{1,000,000}{1282} $.

Convert the total number of troy ounces to grams using the conversion factor: 1 troy ounce = 31.1035 grams. Calculate the total mass in grams: $ \text{Total mass in grams} = \text{Total troy ounces} \times 31.1035 $.

Recall that the density of gold is approximately 19.32 g/cm³. Use this density to find the volume of the gold cube. The formula for volume based on mass and density is: $ \text{Volume} = \frac{\text{Total mass in grams}}{19.32} $.

Since the gold is in the shape of a cube, the volume of the cube is given by $ V = a^3 $, where $ a $ is the side length of the cube. Solve for $ a $ by taking the cube root of the volume: $ a = \sqrt[3]{\text{Volume}} $.

The side length $ a $ represents the height of the cube. Therefore, the height of your million-dollar gold cube is equal to the side length $ a $ calculated in the previous step.

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주요 개념

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Density of Gold

Density is a measure of mass per unit volume, and for gold, it is approximately 19.32 g/cm³. Understanding density is crucial for calculating the volume of gold needed to form a cube worth a million dollars. By knowing the density, you can relate the mass of gold to its volume, which is essential for determining the cube's dimensions.

Conversion of Currency to Mass

To find out how much gold you can buy with a million dollars, you need to convert the currency into mass. This involves using the current price of gold per troy ounce and converting it to grams. By dividing the total money by the price per gram, you can determine the total mass of gold you can purchase, which is necessary for calculating the cube's size.

Volume Calculation of a Cube

The volume of a cube is calculated using the formula V = a³, where 'a' is the length of one side. Once the mass and density of gold are known, you can calculate the volume of the gold cube. From the volume, you can derive the side length, which gives the height of the cube. This concept is essential for translating the mass of gold into a tangible three-dimensional object.