calculate the height from the formula of volume

To find the thickness (height) of the gold leaf sheet using the formula for volume, we can rearrange the volume formula. Let's work through the calculations step-by-step.

### 1. **Formula for Volume:**

The volume (\(V\)) of a rectangular prism (or in this case, a sheet of gold leaf) is given by:

\[
V = \text{Length} \times \text{Width} \times \text{Height}
\]

Where:
- \(V\) is the volume.
- \(\text{Length}\) and \(\text{Width}\) are the dimensions of the gold sheet.
- \(\text{Height}\) is the thickness of the sheet, which we need to find.

### 2. **Given Data:**

- Mass of gold: \(330 \, \text{mg} = 0.33 \, \text{g}\)
- Density of gold: \(19.3 \, \text{g/cm}^3\)
- Length of gold sheet: \(1.88 \, \text{in}\)
- Width of gold sheet: \(3.45 \, \text{in}\)

### 3. **Calculate Volume of Gold:**

Using the density formula, we can find the volume:

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

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

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

### 4. **Convert Length and Width to Centimeters:**

- Length: 

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

- Width: 

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

### 5. **Use the Volume Formula to Find Height (Thickness):**

\[
V = \text{Length} \times \text{Width} \times \text{Height}
\]

Rearranging to solve for height (\(h\)):

\[
h = \frac{V}{\text{Length} \times \text{Width}}
\]

Substituting the known values:

\[
h = \frac{0.0171 \, \text{cm}^3}{4.7752 \, \text{cm} \times 8.763 \, \text{cm}}
\]

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

\[
h \approx 0.0004085 \, \text{cm}
\]

### 6. **Convert Thickness to Meters:**

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

\[
h \approx 0.0004085 \, \text{cm} \times 0.01 \, \text{m/cm}
\]

\[
h \approx 4.085 \times 10^{-6} \, \text{m}
\]

### **Final Answer:**

The average thickness of the gold leaf sheet is approximately \(4.085 \times 10^{-6} \, \text{meters}\).

---

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How would the thickness change if the mass of gold were doubled?
2. What is the volume of a gold cube with a side length of 1 cm?
3. If a piece of gold leaf has a thickness of \(5 \times 10^{-6}\) meters, what would its mass be if it has the same length and width?
4. What would be the area of the gold leaf if the thickness were \(1 \times 10^{-5}\) meters?
5. How does the density of gold affect its use in thin sheets like gold leaf?

**Tip:** Always make sure to convert units consistently when solving problems involving volume, area, and density!

Learn how to calculate the thickness (height) of a gold leaf sheet using the volume formula. This problem involves applying the volume formula for a rectangular prism, converting units from inches to centimeters, and using density calculations. Suitable for advanced high school students and anyone interested in physics and material science.

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