A hemispherical bowl is made of steel, 0.5 cm thick. The inside radius of the bowl is 4 cm. Find the volume of steel used in making the bowl.

Answer

To find the volume of steel used in making the hemispherical bowl, we will follow these steps:

### Step 1: Identify the dimensions of the bowl
- The inside radius of the bowl (r1) = 4 cm
- The thickness of the bowl = 0.5 cm
- Therefore, the outside radius of the bowl (r2) = inside radius + thickness = 4 cm + 0.5 cm = 4.5 cm

### Step 2: Calculate the volume of the outer hemisphere
The formula for the volume of a hemisphere is given by:
\[ V = \frac{2}{3} \pi r^3 \]
Using the outside radius (r2 = 4.5 cm):
\[ V_{outer} = \frac{2}{3} \pi (4.5)^3 \]

### Step 3: Calculate the volume of the inner hemisphere
Using the inside radius (r1 = 4 cm):
\[ V_{inner} = \frac{2}{3} \pi (4)^3 \]

### Step 4: Calculate the volume of steel used
The volume of steel used in making the bowl is the difference between the volume of the outer hemisphere and the volume of the inner hemisphere:
\[ V_{steel} = V_{outer} - V_{inner} \]

### Step 5: Substitute the values and calculate
1. Calculate \( (4.5)^3 \):
\[ (4.5)^3 = 91.125 \]
So,
\[ V_{outer} = \frac{2}{3} \pi (91.125) = \frac{182.25}{3} \pi \approx 60.75 \pi \text{ cm}^3 \]

2. Calculate \( (4)^3 \):
\[ (4)^3 = 64 \]
So,
\[ V_{inner} = \frac{2}{3} \pi (64) = \frac{128}{3} \pi \approx 42.67 \pi \text{ cm}^3 \]

3. Now, calculate the volume of steel:
\[ V_{steel} = V_{outer} - V_{inner} = (60.75 \pi - 42.67 \pi) \text{ cm}^3 = 18.08 \pi \text{ cm}^3 \]

4. Using \( \pi \approx 3.14 \):
\[ V_{steel} \approx 18.08 \times 3.14 \approx 56.67 \text{ cm}^3 \]

### Final Answer:
The volume of steel used in making the bowl is approximately \( 56.67 \text{ cm}^3 \).

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