The difference between the outside and inside surfaces of a cylindrical metal pipe 14 cm long is `44" cm"^(2)`. It the pipe is made of `99" cm"^(3)` of metal, find the sum of the inner and outer radii of the pipe ?

Options

A. `4.5` cm
B. `3.5` cm
C. 2 cm
D. `5.5` cm

Answer

To solve the problem step by step, we will use the information given about the cylindrical metal pipe.

### Step 1: Understand the Given Information
- Length of the pipe (height, H) = 14 cm
- Difference between the outside and inside surface area = 44 cm²
- Volume of the metal used = 99 cm³

Let:
- Inner radius = R1
- Outer radius = R2

### Step 2: Write the Formula for Curved Surface Area
The curved surface area (CSA) of a cylinder is given by the formula:
\[ \text{CSA} = 2\pi r h \]
For the inner and outer surfaces, we have:
- Inner CSA = \( 2\pi R1 H \)
- Outer CSA = \( 2\pi R2 H \)

### Step 3: Set Up the Equation for Surface Area Difference
According to the problem, the difference between the outer and inner surface areas is:
\[ 2\pi R2 H - 2\pi R1 H = 44 \]
Factoring out \( 2\pi H \):
\[ 2\pi H (R2 - R1) = 44 \]

### Step 4: Substitute Known Values
Substituting H = 14 cm:
\[ 2\pi (14) (R2 - R1) = 44 \]
\[ 28\pi (R2 - R1) = 44 \]
Dividing both sides by \( 28\pi \):
\[ R2 - R1 = \frac{44}{28\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ R2 - R1 = \frac{44}{28 \times \frac{22}{7}} = \frac{44 \times 7}{28 \times 22} = \frac{308}{616} = \frac{1}{2} \text{ cm} \]

### Step 5: Write the Volume Equation
The volume of the pipe can be expressed as the difference between the volume of the outer cylinder and the inner cylinder:
\[ V = \pi R2^2 H - \pi R1^2 H \]
Factoring out \( \pi H \):
\[ V = \pi H (R2^2 - R1^2) \]
Substituting V = 99 cm³ and H = 14 cm:
\[ 99 = \pi (14) (R2^2 - R1^2) \]

### Step 6: Substitute the Value of π
Using \( \pi \approx \frac{22}{7} \):
\[ 99 = \frac{22}{7} (14) (R2^2 - R1^2) \]
\[ 99 = \frac{308}{7} (R2^2 - R1^2) \]
Multiplying both sides by 7:
\[ 693 = 308 (R2^2 - R1^2) \]
Dividing both sides by 308:
\[ R2^2 - R1^2 = \frac{693}{308} = \frac{99}{44} = \frac{9}{4} \]

### Step 7: Use the Difference of Squares
Using the identity \( a^2 - b^2 = (a - b)(a + b) \):
\[ R2^2 - R1^2 = (R2 - R1)(R2 + R1) \]
Substituting \( R2 - R1 = \frac{1}{2} \):
\[ \frac{9}{4} = \frac{1}{2} (R2 + R1) \]
Multiplying both sides by 2:
\[ \frac{9}{2} = R2 + R1 \]

### Step 8: Conclusion
Thus, the sum of the inner and outer radii of the pipe is:
\[ R1 + R2 = 4.5 \text{ cm} \]