A drainage tile is a cylindrical shell 21 cm long. The inside and outside diameters are 4.5 cm and 5.1 cm, respectively. What is the volume of the clay required for the tile?

Options

A. a. `6.96 tu cm^(3) `
B. b. `6.76 tu cm^(3) `
C. c. `5.76 tu cm^(3) `
D. d. None of these

Answer

To find the volume of clay required for the drainage tile, we will calculate the volume of the outer cylinder and subtract the volume of the inner cylinder. Here’s the step-by-step solution:

### Step 1: Identify the dimensions
- Length (height) of the cylindrical shell, \( h = 21 \) cm
- Inside diameter, \( d_1 = 4.5 \) cm
- Outside diameter, \( d_2 = 5.1 \) cm

### Step 2: Calculate the radii
- Inside radius, \( r_1 = \frac{d_1}{2} = \frac{4.5}{2} = 2.25 \) cm
- Outside radius, \( r_2 = \frac{d_2}{2} = \frac{5.1}{2} = 2.55 \) cm

### Step 3: Calculate the volume of the outer cylinder
The formula for the volume of a cylinder is given by:
\[ V = \pi r^2 h \]
For the outer cylinder:
\[ V_{\text{outer}} = \pi r_2^2 h = \pi (2.55)^2 (21) \]
Calculating \( (2.55)^2 \):
\[ (2.55)^2 = 6.5025 \]
Now substituting back:
\[ V_{\text{outer}} = \pi (6.5025) (21) = 136.5525\pi \text{ cm}^3 \]

### Step 4: Calculate the volume of the inner cylinder
For the inner cylinder:
\[ V_{\text{inner}} = \pi r_1^2 h = \pi (2.25)^2 (21) \]
Calculating \( (2.25)^2 \):
\[ (2.25)^2 = 5.0625 \]
Now substituting back:
\[ V_{\text{inner}} = \pi (5.0625) (21) = 106.3125\pi \text{ cm}^3 \]

### Step 5: Calculate the volume of clay
The volume of clay required is the difference between the outer and inner volumes:
\[ V_{\text{clay}} = V_{\text{outer}} - V_{\text{inner}} = (136.5525\pi - 106.3125\pi) \text{ cm}^3 \]
\[ V_{\text{clay}} = (30.24\pi) \text{ cm}^3 \]

### Step 6: Final calculation
Using \( \pi \approx 3.14 \):
\[ V_{\text{clay}} \approx 30.24 \times 3.14 \approx 94.91 \text{ cm}^3 \]

Thus, the volume of clay required for the drainage tile is approximately \( 94.91 \text{ cm}^3 \).