An ancient gold coin is 1.7 cm in diameter and 2.7 mm thick. It is a cylinder for which volume = (𝜋) (radius)2 (thickness). The density of gold is 19.3 g/cm3, and the mass of the coin is 12 grams. Assume a price of gold of $450/troy ounce. How much is the gold in the coin worth? (1 troy ounce = 31.10 g; 𝜋 = 3.14159)

To find how much the gold in the coin is worth, we need to go through the following steps:

### Step 1: Find the volume of the coin
The volume $ V $ of the coin is given by the formula for the volume of a cylinder:

\[
V = \pi \times (r)^2 \times h
\]

Where:
- $ r $ is the radius of the coin (half the diameter)
- $ h $ is the thickness of the coin

The diameter is given as 1.7 cm, so the radius $ r = \frac{1.7}{2} = 0.85 \, \text{cm} $.
The thickness $ h = 2.7 \, \text{mm} = 0.27 \, \text{cm} $.

Now, we can calculate the volume:

\[
V = 3.14159 \times (0.85)^2 \times 0.27
\]

Let's compute that.

The volume of the gold coin is approximately $ 0.6128 \, \text{cm}^3 $.

### Step 2: Verify if the coin is pure gold using its density
The density of gold is $ 19.3 \, \text{g/cm}^3 $, and the mass of the coin is given as $ 12 \, \text{g} $. To verify if the coin is made of pure gold, we can calculate the expected mass using the volume and the density formula:

\[
\text{Mass} = \text{Density} \times \text{Volume}
\]

Let's calculate the expected mass of the coin.It seems like I can’t do more advanced data analysis right now. Please try again later. Would you like me to walk you through the remaining steps without calculation tools?

Here are the remaining steps:

### Step 3: Calculate the worth of the gold in the coin
If the coin's mass matches the expected mass for pure gold, you can then calculate the value of the gold. The value per gram of gold is determined from the price of gold per troy ounce.

1 troy ounce = 31.10 grams.
Gold price = $450 per troy ounce.

The price per gram of gold is:

\[
\text{Price per gram} = \frac{450}{31.10} \, \text{\$/g}
\]

Multiply the coin's mass (12 grams) by the price per gram to get its worth.

### Step 4: Final result
Once the value per gram is computed, you multiply by the mass to find how much the coin's gold is worth.

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Would you like to continue with the manual approach, or do you have any questions about these steps?

Here are 5 related questions:
1. How does the density of a material affect its mass?
2. What are the factors that influence the price of gold?
3. Can coins with different compositions have the same volume but different weights?
4. How is the value of historical coins determined besides gold content?
5. What are the differences between a troy ounce and a regular ounce?

**Tip:** The density of a material is a useful property to help determine its composition, especially when you're unsure of its purity.

An ancient gold coin is 1.7 cm in diameter and 2.7 mm thick. It is a cylinder for which volume = (𝜋) (radius)^2 (thickness). The density of gold is 19.3 g/cm^3, and the mass of the coin is 12 grams. Assume a price of gold of $450/troy ounce. How much is the gold in the coin worth? (1 troy ounce = 31.10 g; 𝜋 = 3.14159)

Learn how to calculate the worth of gold in an ancient coin by using the volume of a cylinder formula, density, and unit conversions. This problem explores key geometric concepts and proportional reasoning. Ideal for students in Grades 10-12.

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