A 330 mg piece of gold (density = 19.3 gcm3) is hammered into a sheet of gold leaf that has a length of 1.88 in and a width of 3.45 in. What is the average thickness of the sheet in meters? (the thickness is height)

To find the average thickness of the gold leaf sheet in meters, we can follow these steps:

### 1. **Convert the mass of gold to grams:**
   The given mass is already in milligrams (mg), so we first convert it to grams (g).

   \[
   330 \, \text{mg} = \frac{330 \, \text{mg}}{1000} = 0.33 \, \text{g}
   \]

### 2. **Find the volume of the gold:**
   We can use the density formula to find the volume:

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

   Rearranging to find volume:

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

   Given:
   - Mass \( = 0.33 \, \text{g} \)
   - Density \( = 19.3 \, \text{g/cm}^3 \)

   So,

   \[
   \text{Volume} = \frac{0.33 \, \text{g}}{19.3 \, \text{g/cm}^3}
   \]

   \[
   \text{Volume} \approx 0.0171 \, \text{cm}^3
   \]

### 3. **Convert the dimensions of the gold leaf to centimeters:**

   Given dimensions are in inches:
   - Length \( = 1.88 \, \text{in} \)
   - Width \( = 3.45 \, \text{in} \)

   Conversion factor: \( 1 \, \text{in} = 2.54 \, \text{cm} \)

   \[
   \text{Length} = 1.88 \, \text{in} \times 2.54 \, \text{cm/in} \approx 4.7752 \, \text{cm}
   \]

   \[
   \text{Width} = 3.45 \, \text{in} \times 2.54 \, \text{cm/in} \approx 8.763 \, \text{cm}
   \]

### 4. **Calculate the area of the gold leaf:**

   \[
   \text{Area} = \text{Length} \times \text{Width}
   \]

   \[
   \text{Area} = 4.7752 \, \text{cm} \times 8.763 \, \text{cm} \approx 41.85 \, \text{cm}^2
   \]

### 5. **Calculate the thickness of the gold leaf:**

   The thickness \( h \) can be calculated by using the volume formula:

   \[
   \text{Volume} = \text{Area} \times \text{Thickness}
   \]

   Rearranging to solve for thickness:

   \[
   \text{Thickness} = \frac{\text{Volume}}{\text{Area}}
   \]

   \[
   \text{Thickness} = \frac{0.0171 \, \text{cm}^3}{41.85 \, \text{cm}^2}
   \]

   \[
   \text{Thickness} \approx 0.000408 \, \text{cm}
   \]

### 6. **Convert thickness to meters:**

   \[
   1 \, \text{cm} = 0.01 \, \text{m}
   \]

   \[
   \text{Thickness} \approx 0.000408 \, \text{cm} \times 0.01 \, \text{m/cm}
   \]

   \[
   \text{Thickness} \approx 4.08 \times 10^{-6} \, \text{m}
   \]

### **Final Answer:**

The average thickness of the gold leaf sheet is approximately \( 4.08 \times 10^{-6} \, \text{meters} \).

---

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What is the volume of a 1 kg piece of gold?
2. How would the thickness change if the width of the gold leaf were doubled?
3. What is the surface area of the gold leaf in square meters?
4. How much mass would a 10 cm² sheet of gold leaf with the same thickness have?
5. How does the density of gold compare to other metals like silver or copper?

**Tip:** To easily convert units, remember that \( 1 \, \text{cm} = 0.01 \, \text{m} \) and \( 1 \, \text{m} = 100 \, \text{cm} \).

Learn how to calculate the average thickness of a gold leaf sheet hammered from a 330 mg piece of gold. Follow the step-by-step solution involving density, volume, and unit conversions. Find the thickness in meters with detailed math explanations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation