In the video, the water level was 15 milliliters before adding the chain, and 20 milliliters after adding the chain. When I weighed the chain, its mass was 70.6 grams.

I believe the chain is solid metal (not gold plated), but is probably a blend of gold and other metals. I looked up that gold has a density of 19.3 g/ml, while the other metals like silver and copper usually blended with gold have a combined density of about 9.7 g/ml.

Determine what percentage of the chain is gold, by mass, to at least 1 decimal place?

To determine the percentage of gold in the chain by mass, we can use the concept of density. Since we know the water displacement (which gives us the volume of the chain) and the total mass of the chain, we can compare the densities of gold and the other metals to calculate the percentage of gold.

### Step 1: Find the volume of the chain
The water displacement will give us the volume of the chain.

- Initial water level = 15 mL
- Water level after adding the chain = 20 mL
- Volume of the chain \( V_{\text{chain}} \) = Final water level - Initial water level
\[
V_{\text{chain}} = 20 \, \text{mL} - 15 \, \text{mL} = 5 \, \text{mL}
\]
So, the volume of the chain is \( 5 \, \text{mL} \).

### Step 2: Use the total mass of the chain
The total mass of the chain is given as 70.6 grams.

### Step 3: Define variables
Let:
- \( x \) = fraction of the chain that is gold (by volume).
- \( 1 - x \) = fraction of the chain that is made of other metals (by volume).

The density of the chain is a weighted average of the densities of gold and the other metals. The equation for total mass is:

\[
\text{Total Mass} = (\text{Volume of gold} \times \text{Density of gold}) + (\text{Volume of other metals} \times \text{Density of other metals})
\]

Substitute the values for density and volume:

\[
70.6 \, \text{g} = (x \times 5 \, \text{mL} \times 19.3 \, \text{g/mL}) + ((1 - x) \times 5 \, \text{mL} \times 9.7 \, \text{g/mL})
\]

### Step 4: Solve for \( x \)
Now let's solve this equation.

\[
70.6 = (x \times 5 \times 19.3) + ((1 - x) \times 5 \times 9.7)
\]
\[
70.6 = (96.5x) + (48.5 - 48.5x)
\]
\[
70.6 = 96.5x + 48.5 - 48.5x
\]
\[
70.6 = 48x + 48.5
\]
\[
70.6 - 48.5 = 48x
\]
\[
22.1 = 48x
\]
\[
x = \frac{22.1}{48} \approx 0.4604
\]

So, the fraction of the chain's volume that is gold is approximately \( 0.4604 \), or 46.04%.

### Step 5: Calculate the percentage by mass
To find the percentage of the chain's mass that is gold, we multiply the volume fraction by the ratio of densities:

\[
\text{Percentage of gold by mass} = x \times \frac{\text{Density of gold}}{\text{Density of chain}}
\]
The density of the chain is the total mass divided by the total volume:

\[
\text{Density of chain} = \frac{70.6 \, \text{g}}{5 \, \text{mL}} = 14.12 \, \text{g/mL}
\]

Now, calculate the percentage of gold by mass:

\[
\text{Percentage of gold by mass} = 0.4604 \times \frac{19.3}{14.12} \approx 0.4604 \times 1.366 \approx 0.6289
\]

Thus, approximately 62.89% of the chain's mass is gold.

### Final Answer:
The percentage of the chain that is gold, by mass, is approximately **62.9%**.

---

Would you like any further details on the process, or would you like to go over more related examples?

Here are some related questions:
1. How is density related to volume and mass in practical applications?
2. Could you use this method to determine the composition of any object?
3. What other methods can be used to find the purity of gold in a metal object?
4. How do the properties of gold differ from those of the blended metals?
5. Can the accuracy of this measurement be improved with other instruments?

**Tip:** When using displacement to measure volume, ensure the object doesn't dissolve or react with the water to maintain accurate results.

In the video, the water level was 15 milliliters before adding the chain, and 20 milliliters after adding the chain. When I weighed the chain, its mass was 70.6 grams. I believe the chain is solid metal (not gold plated), but is probably a blend of gold and other metals. I looked up that gold has a density of 19.3 g/ml, while the other metals like silver and copper usually blended with gold have a combined density of about 9.7 g/ml. Determine what percentage of the chain is gold, by mass, to at least 1 decimal place?

Learn how to determine the percentage of gold in a metal chain using the principles of density, weighted averages, and algebra. This step-by-step solution covers the calculation of the chain's volume from water displacement and mass. Ideal for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation