The ratio of curved surface area and volume of a cylinder is 1 : 7. The ratio of total surface area and volume is 187 : 770. What is the respective ratio of its base radius and height?

Options

A. `5 : 8`
B. ` 4 : 9`
C. ` 3 : 7`
D. `7 : 10`

Answer

To solve the problem step by step, we need to find the respective ratio of the base radius (r) and height (h) of a cylinder given certain ratios of its surface areas and volume.

### Step 1: Understand the given ratios
We are given:
1. The ratio of curved surface area (CSA) to volume (V) of a cylinder is 1:7.
2. The ratio of total surface area (TSA) to volume (V) is 187:770.

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

From the ratio of CSA to V:
\[ \frac{CSA}{V} = \frac{2\pi rh}{\pi r^2 h} = \frac{2}{r} = \frac{1}{7} \]
This gives us the equation:
\[ 2 = \frac{r}{7} \implies r = 14 \]

### Step 3: Analyze the total surface area
The formula for total surface area (TSA) of a cylinder is:
\[ TSA = 2\pi rh + 2\pi r^2 \]
Using the volume formula:
\[ V = \pi r^2 h \]
From the ratio of TSA to V:
\[ \frac{TSA}{V} = \frac{2\pi rh + 2\pi r^2}{\pi r^2 h} = \frac{2(r + \frac{r^2}{h})}{r^2} = \frac{187}{770} \]

### Step 4: Simplify the TSA equation
Cancelling \(\pi\) from the numerator and denominator:
\[ \frac{2(r + \frac{r^2}{h})}{r^2} = \frac{187}{770} \]
Cross-multiplying gives:
\[ 2(r + \frac{r^2}{h}) \cdot 770 = 187 \cdot r^2 \]
Substituting \(r = 14\):
\[ 2(14 + \frac{14^2}{h}) \cdot 770 = 187 \cdot 14^2 \]

### Step 5: Solve for height (h)
Calculating \(14^2 = 196\):
\[ 2(14 + \frac{196}{h}) \cdot 770 = 187 \cdot 196 \]
Calculating \(187 \cdot 196 = 36632\):
\[ 2(14 + \frac{196}{h}) \cdot 770 = 36632 \]
Dividing both sides by 1540 (which is \(2 \cdot 770\)):
\[ 14 + \frac{196}{h} = \frac{36632}{1540} = 23.8 \]
Thus:
\[ \frac{196}{h} = 23.8 - 14 = 9.8 \]
This gives:
\[ h = \frac{196}{9.8} = 20 \]

### Step 6: Find the ratio of radius to height
Now we have:
- \(r = 14\)
- \(h = 20\)

The ratio of radius to height is:
\[ \frac{r}{h} = \frac{14}{20} = \frac{7}{10} \]

### Final Answer
The respective ratio of the base radius to height is \(7:10\).