Gold can be hammered into extremely thin sheets called gold leaf. An architect wants to cover a 30 m x 25 m ceiling with gold leaf that is twelve-millionths of a centimeter thick. The density of gold is 19.32 g/cm³, and gold costs $1654 per troy ounce (1 troy ounce = 31.1034768 g). How much will it cost the architect to buy the necessary gold?

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Calculate the volume of gold needed by multiplying the area of the ceiling by the thickness of the gold leaf. Use the formula: $ \text{Volume} = \text{Area} \times \text{Thickness} $.

Convert the volume from cubic centimeters to cubic meters, if necessary, to match the units of the area and thickness.

Use the density of gold to find the mass of gold required. Use the formula: $ \text{Mass} = \text{Density} \times \text{Volume} $.

Convert the mass of gold from grams to troy ounces using the conversion factor: 1 troy ounce = 31.1034768 g.

Calculate the cost by multiplying the number of troy ounces by the cost per troy ounce ($1654).

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Key Concepts

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

Volume and Thickness Calculation

To determine the amount of gold needed, one must calculate the volume of gold leaf required to cover the ceiling. This involves multiplying the area of the ceiling (30 m x 25 m) by the thickness of the gold leaf, converted into centimeters. Understanding how to manipulate units and perform volume calculations is essential for solving the problem.

Density and Mass Relationship

Density is defined as mass per unit volume, and it plays a crucial role in converting the calculated volume of gold leaf into mass. The density of gold (19.32 g/cm³) allows us to find the mass by multiplying the volume by the density. This relationship is fundamental in determining how much gold is needed in grams before converting it to troy ounces for cost calculation.

Cost Calculation and Unit Conversion

Once the mass of gold is determined, the next step is to calculate the cost based on the price per troy ounce. This requires converting grams to troy ounces using the conversion factor (1 troy ounce = 31.1034768 g). Understanding how to perform these conversions and apply the cost per unit is essential for arriving at the final cost for the architect.

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