Gold can be hammered into extremely thin sheets called gold...

Gold can be hammered into extremely thin sheets called gold leaf. An architect wants to cover a $30 \mathrm{~m} \times 25 \mathrm{~m}$ ceiling with gold leaf that is twelve-millionths of a centimeter thick. The density of gold is $19.32 \mathrm{~g} / \mathrm{cm}^{3}$, and gold costs $\$ 1654$ per troy ounce ( 1 troy ounce $=31.1034768 \mathrm{~g}$ ). How much will it cost the architect to buy the necessary gold?

Key Concepts

Mass Conversion

Mass conversion refers to changing the measurement of mass from one unit system to another, such as from grams to ounces. This is important when the pricing or other factors are provided in a unit different from that used in the initial calculation.

Density

Density is the property that relates mass to volume, typically expressed as mass per unit volume. By using density, one can calculate the mass of a substance when its volume is known, which is fundamental when determining material quantities based on physical dimensions.

Cost Calculation

Cost calculation involves determining the total expense by multiplying the quantity of material (in appropriate units) by the cost per unit. This step requires careful integration of all previous calculations to yield a final monetary value.

Volume Calculation

This concept involves determining the volume of a material by multiplying its surface area by its thickness. In problems where a thin layer is applied over a large area, calculating the volume accurately is essential to assess how much raw material is needed.

Unit Conversion

Unit conversion is critical in problems where different measurement systems or scales are used. It involves converting measurements such as length, area, or volume from one unit to another, ensuring that all quantities are in compatible units for accurate calculation.

Transcript

00:01 Okay, so for this question, i've already written the known amounts that were given us in the equation.

00:05 So they're trying to cover a 30 meter by 25 meter ceiling with gold leaf.

00:12 That is 12 millionths of a centimeter thick.

00:16 I'm wondering how much it would cost.

00:21 So remember that the density is mass divided by the volume.

00:29 So if we rearrange this equation, mass is going to be equal to the density times the volume.

00:37 So we can use the area that needs to be covered to figure out the volume.

00:44 Then we are already given the density.

00:48 So from those two we can figure out the mass.

00:51 Once we know the mass or the weight in grams of the amount of gold that we need, then we can use these two conversions here to figure out how much that amount of gold would cost.

01:06 So first, let's figure out the volume.

01:11 So if you'll notice, right, the area is 30 meters by 25 meters, but we want it to be in centimeters cubed.

01:18 So let's go ahead and convert our meters into centimeters.

01:22 So 30 meters times, there's 100 centimeters, and 1 meter.

01:30 So that's gonna be 3 ,000 centimeters.

01:38 And then same with the 25, right, we take it times 100 and it's gonna be 2 ,500 centimeters.

01:49 So now to get the volume, right, it's gonna be the height plus the width, length plus the width...

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