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Ch 12: Fluid Mechanics
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 12: Fluid MechanicsProblem 4
Chapter 12, Problem 4

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?

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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Related Practice
Textbook Question

Oceans on Mars. Scientists have found evidence that Mars may once have had an ocean 0.500 km deep. The acceleration due to gravity on Mars is 3.71 m/s2. (a) What would be the gauge pressure at the bottom of such an ocean, assuming it was freshwater? (b) To what depth would you need to go in the earth's ocean to experience the same gauge pressure?

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Textbook Question

In intravenous feeding, a needle is inserted in a vein in the patient's arm and a tube leads from the needle to a reservoir of fluid (density 1050 kg/m3) located at height h above the arm. The top of the reservoir is open to the air. If the gauge pressure inside the vein is 5980 Pa, what is the minimum value of h that allows fluid to enter the vein? Assume the needle diameter is large enough that you can ignore the viscosity of the liquid.

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Textbook Question

On a part-time job, you are asked to bring a cylindrical iron rod of length 85.8 cm and diameter 2.85 cm from a storage room to a machinist. Will you need a cart? (To answer, calculate the weight of the rod.)

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Textbook Question

A cube 5.0 cm on each side is made of a metal alloy. After you drill a cylindrical hole 2.0 cm in diameter all the way through and perpendicular to one face, you find that the cube weighs 6.30 N. What is the density of this metal?

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Textbook Question

A barrel contains a 0.120-m layer of oil floating on water that is 0.250 m deep. The density of the oil is 600 kg/m3. (a) What is the gauge pressure at the oil–water interface? (b) What is the gauge pressure at the bottom of the barrel?