Calculating Metallic Cylindrical Pipe Length
This problem involves finding the length of a metallic cylindrical pipe given its volume, external radius, and thickness. A metallic pipe is essentially a hollow cylinder.
Given Information
- Volume of the pipe, $V = 1232\text{ cm}^3$
- External Radius, $R = 8 \text{ cm}$
- Thickness of the metal, $t = 2 \text{ cm}$
- Length of the pipe, $h$ (to be determined)
Understanding the Formula for Hollow Cylinder Volume
The volume of a hollow cylinder is the difference between the volume of the outer cylinder and the volume of the inner cylinder. The formula is:
$$ V = \pi (R^2 - r^2)h $$Where:
- $V$ is the volume
- $R$ is the external radius
- $r$ is the internal radius
- $h$ is the length (or height) of the cylinder
We need to find the internal radius ($r$) first.
Step 1: Calculate the Internal Radius (r)
The internal radius is found by subtracting the thickness from the external radius:
$$ r = R - t $$Substituting the given values:
$$ r = 8 \text{ cm} - 2 \text{ cm} = 6 \text{ cm} $$So, the internal radius is $6 \text{ cm}$.
Step 2: Use the Volume Formula to Find the Length (h)
Now, substitute the known values ($V$, $R$, and $r$) into the volume formula:
$$ 1232 = \pi (8^2 - 6^2)h $$Calculate the squares:
$$ 1232 = \pi (64 - 36)h $$Subtract the values inside the parentheses:
$$ 1232 = \pi (28)h $$Now, substitute the value of $\pi$ as $\frac{22}{7}$:
$$ 1232 = \left(\frac{22}{7}\right) \times 28 \times h $$Simplify the expression:
$$ 1232 = 22 \times \left(\frac{28}{7}\right) \times h $$ $$ 1232 = 22 \times 4 \times h $$ $$ 1232 = 88h $$Step 3: Solve for the Length (h)
To find $h$, divide the volume by 88:
$$ h = \frac{1232}{88} $$Performing the division:
$$ h = 14 $$Therefore, the length of the pipe is $14 \text{ cm}$.
Final Answer Summary
The calculated length of the metallic cylindrical pipe is $14 \text{ cm}$. This matches one of the options provided.
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