The ratio of the numerical values of curved surface area to the volume of the right circular cylinder is 1:7. If the ratio of the diameter to the height of the cylinder is 7:5, then what is the total surface area of the cylinder? (in `m^2`)

Options

A. 2992
B. 3172
C. 2882
D. 3576

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given ratios
We know that the ratio of the curved surface area (CSA) to the volume (V) of the cylinder is 1:7. We also know that the ratio of the diameter (d) to the height (h) of the cylinder is 7:5.

### Step 2: Define the variables
Let:
- The radius of the cylinder be \( r \) meters.
- The height of the cylinder be \( h \) meters.

From the diameter to height ratio:
\[ \frac{d}{h} = \frac{7}{5} \]
Since the diameter \( d = 2r \), we can write:
\[ \frac{2r}{h} = \frac{7}{5} \]

### Step 3: Express height in terms of radius
From the above equation, we can express \( h \) in terms of \( r \):
\[ 2r \cdot 5 = 7h \implies h = \frac{10r}{7} \]

### Step 4: Write the formulas for CSA and Volume
The formulas for the curved surface area and volume of a cylinder are:
- Curved Surface Area (CSA) = \( 2\pi rh \)
- Volume (V) = \( \pi r^2 h \)

### Step 5: Set up the ratio of CSA to Volume
Using the ratio given in the problem:
\[ \frac{CSA}{V} = \frac{1}{7} \]
Substituting the formulas:
\[ \frac{2\pi rh}{\pi r^2 h} = \frac{1}{7} \]
This simplifies to:
\[ \frac{2}{r} = \frac{1}{7} \]

### Step 6: Solve for radius \( r \)
Cross-multiplying gives:
\[ 2 \cdot 7 = r \implies r = 14 \text{ meters} \]

### Step 7: Find height \( h \)
Now substituting \( r \) back into the equation for \( h \):
\[ h = \frac{10r}{7} = \frac{10 \cdot 14}{7} = 20 \text{ meters} \]

### Step 8: Calculate the Total Surface Area (TSA)
The formula for the total surface area of a cylinder is:
\[ TSA = 2\pi r (r + h) \]
Substituting the values of \( r \) and \( h \):
\[ TSA = 2\pi \cdot 14 \cdot (14 + 20) = 2\pi \cdot 14 \cdot 34 \]
Calculating:
\[ TSA = 28\pi \cdot 34 = 952\pi \]
Using \( \pi \approx \frac{22}{7} \):
\[ TSA = 952 \cdot \frac{22}{7} = 2992 \text{ m}^2 \]

### Final Answer
The total surface area of the cylinder is \( 2992 \text{ m}^2 \).

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