A hollow garden roller of 42cm diameter and length 252cm is made of cast iron 3cm thick. Find the volume of iron used in the roller.

To calculate the volume of iron used in the hollow garden roller, we need to find the volume of the roller as a solid cylinder first, and then subtract the volume of the hollow part (empty space) from it.

### Step-by-step process:

1. **Given dataTo calculate the volume of iron used in the hollow garden roller, we need to find the volume of the roller as a solid cylinder first, and then subtract the volume of the hollow part (empty space) from it.

### Step-by-step process:

1. **Given **
   - Outer diameter of the roller = 42 cm
   - Thickness of the roller = 3 cm
   - Length of the roller = 252 cm

2. **Calculate the inner diameter:**
   The inner diameter is the outer diameter minus twice the thickness (because the thickness is on both sides of the roller):
   \[
   \text{Inner diameter} = 42 \, \text{cm} - 2 \times 3 \, \text{cm} = 36 \, \text{cm}
   \]

3. **Calculate the outer and inner radii:**
   - Outer radius \( R \) = \( \frac{42}{2} = 21 \, \text{cm} \)
   - Inner radius \( r \) = \( \frac{36}{2} = 18 \, \text{cm} \)

4. **Formula for the volume of a cylinder:**
   The volume \( V \) of a cylinder is given by the formula:
   \[
   V = \pi r^2 h
   \]
   where \( r \) is the radius and \( h \) is the height (length) of the cylinder.

5. **Calculate the volume of the outer cylinder (solid):**
   \[
   V_{\text{outer}} = \pi R^2 h = \pi (21^2) \times 252
   \]

6. **Calculate the volume of the inner hollow part:**
   \[
   V_{\text{inner}} = \pi r^2 h = \pi (18^2) \times 252
   \]

7. **Find the volume of the iron used:**
   The volume of iron used in the roller is the difference between the outer volume and the inner volume:
   \[
   V_{\text{iron}} = V_{\text{outer}} - V_{\text{inner}} = \pi \times (21^2 - 18^2) \times 252
   \]
   Simplify the expression:
   \[
   V_{\text{iron}} = \pi \times (441 - 324) \times 252 = \pi \times 117 \times 252
   \]
   \[
   V_{\text{iron}} = \pi \times 29574
   \]
   Approximate the value of \( \pi \) (3.1416):
   \[
   V_{\text{iron}} = 3.1416 \times 29574 \approx 92883.55 \, \text{cm}^3
   \]

Thus, the volume of iron used in the roller is approximately **92,883.55 cubic centimeters**.

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Do you want more details or have any questions about this?

Here are 5 related questions:
1. How does the volume change if the thickness of the roller increases?
2. How would the calculation differ if the roller were filled with material instead of hollow?
3. What is the weight of the roller if the density of cast iron is known?
4. How does the diameter affect the amount of material used in construction?
5. Can you calculate the surface area of the roller?

**Tip:** Always subtract the inner hollow volume from the outer volume for hollow objects to find the material volume used.

Learn how to calculate the volume of iron used in a hollow garden roller with a diameter of 42 cm, length of 252 cm, and 3 cm thickness. This solution uses geometry concepts and cylindrical volume formulas to find the result, making it perfect for students in Grades 9-11.

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