Ch.1 - Introduction: Matter, Energy, and Measurement

Problem 67

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Chapter 1, Problem 67

Gold can be hammered into extremely thin sheets called gold leaf. An architect wants to cover a 100 ft x 82 ft ceiling with gold leaf that is five-millionths of an inch thick. The density of gold is 19.32 g/cm³. If gold costs $1768 per troy ounce (1 troy ounce = 31.10348 g), how much will it cost the architect to buy the necessary gold?

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Convert the dimensions of the ceiling from feet to inches, since the thickness of the gold leaf is given in inches.

Calculate the volume of gold needed by multiplying the area of the ceiling (in square inches) by the thickness of the gold leaf (in inches).

Convert the volume from cubic inches to cubic centimeters, as the density of gold is given in g/cm^3.

Use the density of gold to find the mass of gold required by multiplying the volume (in cm^3) by the density (in g/cm^3).

Convert the mass of gold from grams to troy ounces, and then calculate the cost by multiplying the number of troy ounces by the cost per troy ounce.

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Key Concepts

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

Density

Density is defined as mass per unit volume, typically expressed in grams per cubic centimeter (g/cm³). In this context, the density of gold (19.32 g/cm³) is crucial for calculating the mass of gold needed to cover the specified area. Understanding density allows us to convert the volume of gold leaf required into mass, which is essential for determining cost.

Volume Calculation

To find the amount of gold needed, we must calculate the volume of the gold leaf that will cover the ceiling. The volume can be determined by multiplying the area of the ceiling by the thickness of the gold leaf. This step is vital as it provides the necessary volume to convert into mass using the density of gold.

Cost Conversion

The cost of gold is typically given in troy ounces, so converting the mass of gold from grams to troy ounces is necessary for calculating the total cost. This involves using the conversion factor (1 troy ounce = 31.10348 g) to ensure that the final cost reflects the price per troy ounce of gold. Understanding this conversion is essential for accurately determining the total expenditure for the architect.

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Conversion Factors Example

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a. What is the density of tungsten in pounds per cubic foot?

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