The ratio of curved surface area to total surface area of a cylinder is 7:9. Find diameter of cylinder, if volume of cylinder is `88 cm^(3).`

Options

A. 4 cm
B. 2 cm
C. 9 cm
D. 7 cm

Answer

To solve the problem step by step, we will follow the given information and formulas related to the cylinder.

### Step 1: Understand the given information
We know that the ratio of the curved surface area (CSA) to the total surface area (TSA) of a cylinder is 7:9, and the volume of the cylinder is 88 cm³.

### Step 2: Write the formulas
The formulas for the curved surface area and total surface area of a cylinder are:
- Curved Surface Area (CSA) = \(2 \pi r h\)
- Total Surface Area (TSA) = \(2 \pi r (r + h)\)

### Step 3: Set up the ratio
Given the ratio of CSA to TSA:
\[ \frac{CSA}{TSA} = \frac{7}{9} \]
Substituting the formulas:
\[ \frac{2 \pi r h}{2 \pi r (r + h)} = \frac{7}{9} \]
This simplifies to:
\[ \frac{h}{r + h} = \frac{7}{9} \]

### Step 4: Cross-multiply to solve for h
Cross-multiplying gives:
\[ 9h = 7(r + h) \]
Expanding the right side:
\[ 9h = 7r + 7h \]
Rearranging gives:
\[ 9h - 7h = 7r \implies 2h = 7r \]
Thus, we can express \(h\) in terms of \(r\):
\[ h = \frac{7}{2}r \]

### Step 5: Write the volume formula
The volume \(V\) of the cylinder is given by:
\[ V = \pi r^2 h \]
Substituting the volume and the expression for \(h\):
\[ 88 = \pi r^2 \left(\frac{7}{2}r\right) \]
This simplifies to:
\[ 88 = \frac{7}{2} \pi r^3 \]

### Step 6: Solve for \(r^3\)
Rearranging gives:
\[ r^3 = \frac{88 \times 2}{7 \pi} \]
Calculating the right side:
\[ r^3 = \frac{176}{7 \pi} \]

### Step 7: Substitute the value of \(\pi\)
Using \(\pi \approx 22/7\):
\[ r^3 = \frac{176}{7 \times \frac{22}{7}} = \frac{176}{22} = 8 \]

### Step 8: Find \(r\)
Taking the cube root:
\[ r = \sqrt[3]{8} = 2 \text{ cm} \]

### Step 9: Find the diameter
The diameter \(d\) of the cylinder is given by:
\[ d = 2r = 2 \times 2 = 4 \text{ cm} \]

### Final Answer
The diameter of the cylinder is **4 cm**.

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