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 76.8 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?


We can determine the percentage of gold in the chain by using the concept of density and mass displacement.

### Given dataWe can determine the percentage of gold in the chain by using the concept of density and mass displacement.

### Given 
- Initial water volume = 15 mL
- Final water volume = 20 mL
- Volume displacement due to chain = \( 20 \, \text{mL} - 15 \, \text{mL} = 5 \, \text{mL} \)
- Mass of the chain = 76.8 g
- Density of gold = 19.3 g/mL
- Density of blended metals (like silver and copper) = 9.7 g/mL

The key formula here is:
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]
To solve for the percentage of gold in the chain, we assume that the chain is a mixture of gold and blended metals, so we can express the total mass of the chain as the sum of the masses of the gold and the other metals:
\[
M_{\text{chain}} = M_{\text{gold}} + M_{\text{other metals}} = 76.8 \, \text{g}
\]
Where:
\[
M_{\text{gold}} = \text{density of gold} \times \text{volume of gold}, \quad \text{and} \quad M_{\text{other metals}} = \text{density of other metals} \times \text{volume of other metals}
\]

Let \( V_{\text{gold}} \) be the volume of the chain made up of gold, and \( V_{\text{other metals}} \) be the volume of the chain made up of other metals. The total volume displacement is:
\[
V_{\text{gold}} + V_{\text{other metals}} = 5 \, \text{mL}
\]
Using the mass equations:
\[
M_{\text{gold}} = 19.3 \, \text{g/mL} \times V_{\text{gold}}
\]
\[
M_{\text{other metals}} = 9.7 \, \text{g/mL} \times V_{\text{other metals}}
\]
And we know:
\[
M_{\text{gold}} + M_{\text{other metals}} = 76.8 \, \text{g}
\]

Substitute the expressions for \( M_{\text{gold}} \) and \( M_{\text{other metals}} \) into the mass equation:
\[
19.3 \times V_{\text{gold}} + 9.7 \times (5 - V_{\text{gold}}) = 76.8
\]
Simplify and solve for \( V_{\text{gold}} \):
\[
19.3 \times V_{\text{gold}} + 9.7 \times 5 - 9.7 \times V_{\text{gold}} = 76.8
\]
\[
19.3 \times V_{\text{gold}} - 9.7 \times V_{\text{gold}} = 76.8 - 9.7 \times 5
\]
\[
(19.3 - 9.7) \times V_{\text{gold}} = 76.8 - 48.5
\]
\[
9.6 \times V_{\text{gold}} = 28.3
\]
\[
V_{\text{gold}} = \frac{28.3}{9.6} \approx 2.95 \, \text{mL}
\]

Thus, the volume of gold in the chain is approximately 2.95 mL. Now, we can calculate the mass of gold:
\[
M_{\text{gold}} = 19.3 \, \text{g/mL} \times 2.95 \, \text{mL} \approx 56.9 \, \text{g}
\]

Finally, the percentage of gold by mass in the chain is:
\[
\text{Percentage of gold} = \frac{M_{\text{gold}}}{M_{\text{chain}}} \times 100 = \frac{56.9}{76.8} \times 100 \approx 74.1\%
\]

### Conclusion:
The chain is approximately **74.1% gold** by mass.

Do you need any further details or clarifications?

Here are some related questions:
1. What would happen to the result if the density of gold were higher?
2. How would the calculation change if the chain was hollow inside?
3. What are the implications of having a higher density blend of metals?
4. Can you apply the same technique to calculate the percentage composition of other alloys?
5. How would the result change if you used different units for volume or mass?

**Tip:** Always verify the density values when working with different alloys, as they vary based on composition.

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 76.8 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 calculate the percentage of gold in a metal chain using concepts of density and volume displacement. This example explores the relationship between mass, volume, and density to determine the gold content. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation