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Chapter 1: Problem 55

Gold can be hammered into extremely thin sheets called gold leaf. An architect wants to cover a \(100 \mathrm{ft} \times 82 \mathrm{ft}\) ceiling with gold leaf that is five-millionths of an inch thick. The density of gold is \(19.32 \mathrm{~g} / \mathrm{cm}^{3},\) and gold costs \(\$ 953\) per troy ounce \((1\) troy ounce \(=31.1034768 \mathrm{~g}) .\) How much will it cost the architect to buy the necessary gold?

Short Answer

Expert verified
The architect will need to spend $57,133.32 to buy the necessary gold to cover the ceiling.

Step by step solution

01

Determine the Area of the Ceiling

To determine the total volume of gold needed to cover the ceiling, we first need to find the area of the ceiling. The ceiling is given to be \(100\,\mathrm{ft} \times 82\,\mathrm{ft}\). To calculate the area, we'll multiply the length and the width of the ceiling. Area = Length × Width = \(100 \mathrm{ft} \times 82 \mathrm{ft} = 8200 \mathrm{ft}^2\)
02

Convert Area to Square Centimeters

Since we have the density and thickness of the gold leaf given in metric units, we need to convert our area from square feet to square centimeters. We can do this by using the conversion factor of \(1\, \mathrm{ft}^2 = 929.0304 \, \mathrm{cm}^2\). Area (in cm²) = \(8200 \mathrm{ft}^2 \times 929.0304 \mathrm{cm}^2/\mathrm{ft}^2 = 7614648.38 \mathrm{~cm}^2\)
03

Calculate the Volume of Gold Leaf

We will now use the given thickness of the gold leaf to find the volume required to cover the entire ceiling. The thickness of the gold leaf is given as five millionths of an inch, which we need to convert to centimeters. Using the conversion factor of \(1\, \mathrm{inch} = 2.54 \mathrm{cm}\), we can find the thickness in centimeters. Thickness (in cm) = \(5\times10^{-6} \mathrm{inch} \times 2.54 \mathrm{cm}/\mathrm{inch} = 1.27 \times 10^{-5} \mathrm{~cm}\) Now, we can find the volume of gold leaf needed: Volume = Area × Thickness = \(7614648.38 \mathrm{~cm}^2 \times 1.27 \times 10^{-5} \mathrm{~cm} = 96.6460 \mathrm{~cm}^3\)
04

Calculate the Mass of Gold Needed

Given the density of the gold, we will now calculate the mass of gold required to cover the ceiling. We can do this using the formula Mass = Density × Volume. Mass = Density × Volume = \(19.32 \mathrm{~g} /\mathrm{cm}^3 \times 96.6460\mathrm{~cm}^3 = 1868.4867 \mathrm{~g}\)
05

Convert Mass to Troy Ounces and Calculate Cost

Now, we will convert this mass from grams to troy ounces using the given conversion factor of \(1\, \mathrm{troy\, ounce} = 31.1034768 \mathrm{~g}\). Mass (in troy ounces) = \(1868.4867 \mathrm{~g} \times (1 \mathrm{troy\, ounce}/31.1034768 \mathrm{~g}) = 60.0140 \mathrm{troy\, ounces}\) Finally, using the given cost of gold per troy ounce, we can calculate the total cost required. Total Cost = Mass (in troy ounces) × Cost per troy ounce = \(60.0140 \mathrm{troy\, ounces} \times \$953/\mathrm{troy\, ounce} = \$57133.32\) So, it will cost the architect $57,133.32 to buy the necessary gold to cover the ceiling.

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