A cylindrical metallic pipe is 14 cm long. The difference between the outside and inside surface is `44 cm^(2)`. If the pipe is made up of 99 cubic cm of metal, find the outer and inner radii of the pipe.

Answer

To solve the problem of finding the outer and inner radii of a cylindrical metallic pipe, we will follow these steps:

### Step 1: Define Variables
Let:
- \( R \) = outer radius of the pipe (in cm)
- \( r \) = inner radius of the pipe (in cm)
- \( h \) = height of the pipe = 14 cm

### Step 2: Surface Area Equation
We know that the difference between the outside and inside surface area is given as \( 44 \, \text{cm}^2 \). The surface area of a cylinder is given by the formula:
\[ \text{Surface Area} = 2 \pi R h \]
For the outer surface area and inner surface area, we have:
\[ 2 \pi R h - 2 \pi r h = 44 \]
Factoring out \( 2 \pi h \):
\[ 2 \pi h (R - r) = 44 \]
Substituting \( h = 14 \):
\[ 2 \pi (14) (R - r) = 44 \]
This simplifies to:
\[ 28 \pi (R - r) = 44 \]
Dividing both sides by \( 28 \pi \):
\[ R - r = \frac{44}{28 \pi} = \frac{11}{7 \pi} \]

### Step 3: Volume Equation
The volume of the metal used in the pipe is given as \( 99 \, \text{cm}^3 \). The volume of the metallic part can be expressed as:
\[ \text{Volume} = \text{Volume of outer cylinder} - \text{Volume of inner cylinder} \]
Using the formula for the volume of a cylinder:
\[ \pi R^2 h - \pi r^2 h = 99 \]
Factoring out \( \pi h \):
\[ \pi h (R^2 - r^2) = 99 \]
Substituting \( h = 14 \):
\[ \pi (14) (R^2 - r^2) = 99 \]
Dividing both sides by \( 14 \pi \):
\[ R^2 - r^2 = \frac{99}{14 \pi} \]

### Step 4: Using the Identity
We can use the identity \( a^2 - b^2 = (a - b)(a + b) \):
\[ R^2 - r^2 = (R - r)(R + r) \]
Substituting \( R - r = \frac{11}{7 \pi} \):
\[ (R - r)(R + r) = \frac{11}{7 \pi} (R + r) = \frac{99}{14 \pi} \]
Now, we can solve for \( R + r \):
\[ R + r = \frac{99}{14 \pi} \cdot \frac{7 \pi}{11} = \frac{99 \cdot 7}{14 \cdot 11} = \frac{693}{154} = \frac{99}{22} = 4.5 \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( R - r = \frac{11}{7 \pi} \)
2. \( R + r = 4.5 \)

Let’s solve these equations:
Adding the two equations:
\[ (R - r) + (R + r) = \frac{11}{7 \pi} + 4.5 \]
This simplifies to:
\[ 2R = \frac{11}{7 \pi} + 4.5 \]
Now, we can find \( R \):
\[ R = \frac{1}{2} \left(\frac{11}{7 \pi} + 4.5\right) \]

Subtracting the two equations:
\[ (R + r) - (R - r) = 4.5 - \frac{11}{7 \pi} \]
This simplifies to:
\[ 2r = 4.5 - \frac{11}{7 \pi} \]
Now, we can find \( r \):
\[ r = \frac{1}{2} \left(4.5 - \frac{11}{7 \pi}\right) \]

### Step 6: Calculate Values
Now, we can calculate the numerical values for \( R \) and \( r \).

### Final Answer
After calculating, we find:
- Inner radius \( r \approx 2 \, \text{cm} \)
- Outer radius \( R \approx 2.5 \, \text{cm} \)