Determine the number of square feet of wall surface on the inside of a wooden silo that has a capacity of 3,392.7 cubic feet and a height of 30 feet.

To determine the square footage of the wall surface on the inside of the silo, let's go through the calculations step-by-step.

### Step 1: Understanding the Shape and Dimensions
The silo is a cylinder, with:
- Volume \( V = 3,392.7 \, \text{cubic feet} \)
- Height \( h = 30 \, \text{feet} \)

### Step 2: Formula for the Volume of a Cylinder
The volume of a cylinder is given by:
\[
V = \pi r^2 h
\]
where:
- \( V \) is the volume,
- \( r \) is the radius, and
- \( h \) is the height.

### Step 3: Solve for the Radius \( r \)
Rearrange the formula to solve for \( r \):
\[
r^2 = \frac{V}{\pi h}
\]
Substitute the given values:
\[
r^2 = \frac{3,392.7}{\pi \times 30}
\]
\[
r^2 = \frac{3,392.7}{94.2478} \approx 36
\]
\[
r \approx \sqrt{36} = 6 \, \text{feet}
\]

### Step 4: Calculate the Wall Surface Area (Lateral Surface Area)
The formula for the lateral surface area of a cylinder (the wall surface inside the silo) is:
\[
A = 2 \pi r h
\]
Substitute \( r = 6 \) feet and \( h = 30 \) feet:
\[
A = 2 \pi \times 6 \times 30
\]
\[
A = 360 \pi
\]
\[
A \approx 1,130.97 \, \text{square feet}
\]

### Final Answer
The inside wall surface area of the silo is approximately **1,130.97 square feet**.

Would you like more details, or do you have any questions?

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Here are 5 related questions to deepen your understanding:
1. How would the wall surface area change if the height of the silo were doubled?
2. What would be the volume of a silo with the same radius but a different height?
3. How does the volume formula for a cylinder compare to that of a sphere?
4. If the silo had a roof, what would be the total surface area (including top and bottom)?
5. How do different values of radius and height affect the volume of a cylinder?

**Tip:** For cylinders, the wall area is directly proportional to both the radius and the height, so any increase in these measurements will increase the surface area.

Learn how to calculate the inside wall surface area of a cylindrical wooden silo with a capacity of 3,392.7 cubic feet and a height of 30 feet. This detailed solution includes the use of formulas for the volume and surface area of a cylinder, making it suitable for students in Grades 9-11.

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