The total surface area of the cylider is `2640 m^(2)` and the sum of height and radius of base of cylinder is 30 m. What is the ratio of height and radius of the cylinder ?

Options

A. `7:9`
B. `9:7`
C. `8:7`
D. `3:7`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the formulas and given information
The total surface area (TSA) of a cylinder is given by the formula:
\[ \text{TSA} = 2\pi r (r + h) \]
where \( r \) is the radius and \( h \) is the height of the cylinder. We are given:
- Total Surface Area (TSA) = 2640 m²
- The sum of height and radius \( (h + r) = 30 \) m

### Step 2: Substitute the known values into the TSA formula
We can substitute the known values into the TSA formula:
\[ 2640 = 2\pi r (r + h) \]
Using \( \pi \approx \frac{22}{7} \), we get:
\[ 2640 = 2 \times \frac{22}{7} r (r + h) \]

### Step 3: Substitute \( h \) in terms of \( r \)
From the equation \( h + r = 30 \), we can express \( h \) as:
\[ h = 30 - r \]
Now substitute \( h \) in the TSA equation:
\[ 2640 = 2 \times \frac{22}{7} r (r + (30 - r)) \]
This simplifies to:
\[ 2640 = 2 \times \frac{22}{7} r \times 30 \]

### Step 4: Simplify the equation
Now simplify the equation:
\[ 2640 = \frac{44}{7} \times 30r \]
\[ 2640 = \frac{1320r}{7} \]
To eliminate the fraction, multiply both sides by 7:
\[ 2640 \times 7 = 1320r \]
\[ 18480 = 1320r \]

### Step 5: Solve for \( r \)
Now, divide both sides by 1320 to find \( r \):
\[ r = \frac{18480}{1320} \]
Calculating this gives:
\[ r = 14 \text{ m} \]

### Step 6: Find \( h \)
Now that we have \( r \), we can find \( h \) using \( h = 30 - r \):
\[ h = 30 - 14 = 16 \text{ m} \]

### Step 7: Find the ratio of height to radius
Now we can find the ratio of height to radius:
\[ \text{Ratio} = \frac{h}{r} = \frac{16}{14} \]
This can be simplified:
\[ \text{Ratio} = \frac{8}{7} \]

### Final Answer
The ratio of height to radius of the cylinder is \( 8:7 \).