how thick is a wall if three layers of 5/8 inch sheetrock are installed on one side of a 5 7/8 inch metal stud and two layers of 5/8inch sheetrock  are installed on top of one layer of 7/8inch plywood on the other side of the stud

To determine the total thickness of the wall, we need to calculate the thickness of each layer on both sides of the metal stud and then add them together.

### Thickness Calculation for Each Side:

1. **One side of the stud:**
   - Three layers of 5/8 inch sheetrock:
     \[
     3 \times \frac{5}{8} = \frac{15}{8} \text{ inches} = 1.875 \text{ inches}
     \]

2. **Other side of the stud:**
   - One layer of 7/8 inch plywood:
     \[
     \frac{7}{8} = 0.875 \text{ inches}
     \]
   - Two layers of 5/8 inch sheetrock:
     \[
     2 \times \frac{5}{8} = \frac{10}{8} \text{ inches} = 1.25 \text{ inches}
     \]

### Adding up the thickness:

- **Total thickness of layers on one side:**  
  \(1.875\) inches (three layers of sheetrock)

- **Total thickness of layers on the other side:**  
  \(0.875\) inches (one layer of plywood) + \(1.25\) inches (two layers of sheetrock)  
  \[
  0.875 + 1.25 = 2.125 \text{ inches}
  \]

- **Metal stud thickness:**  
  \(5 \frac{7}{8}\) inches = \(5.875\) inches

### Total Wall Thickness:

Now, add the thickness of all the layers and the metal stud:

\[
\text{Total thickness} = 1.875 + 5.875 + 2.125
\]

\[
\text{Total thickness} = 9.875 \text{ inches}
\]

So, the total thickness of the wall is **9.875 inches**.

Would you like any further details or have any questions?

### Related Questions:

1. How would the thickness change if an additional layer of sheetrock is added to each side?
2. What is the effect on sound insulation when more layers of sheetrock are added?
3. How would the total thickness differ if the metal stud was 6 inches instead?
4. What materials could be added to increase the thermal insulation of the wall?
5. How would changing the plywood thickness to 3/4 inch affect the total wall thickness?

### Tip:
Always double-check your measurements and calculations to ensure accuracy, especially when dealing with construction materials.

Learn how to calculate the total thickness of a wall with multiple layers of sheetrock, plywood, and a metal stud. This problem involves measurements and geometry, focusing on adding up the thickness of each layer on both sides of the metal stud. Suitable for advanced high school students and construction professionals.

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