For the following questions answer them individually
The time required for a boy to travel along the external and internal boundaries of a circular path are in the ratio 20: 19. If the width of the path be 5 metres, the internal diameter is:

Options

A. 195 metres
B. 192 metres
C. 180 metres
D. 190 metres

Answer

To solve the problem step-by-step, we will follow the logical reasoning laid out in the video transcript.

### Step-by-Step Solution:

1. **Understand the Problem**: We have a circular path with an external and internal boundary. The time taken to travel along these boundaries is in the ratio of 20:19. The width of the path is given as 5 meters. We need to find the internal diameter of the circular path.

2. **Define Variables**:
- Let the internal radius be \( r_1 \).
- The external radius will then be \( r_2 = r_1 + 5 \) (since the width of the path is 5 meters).

3. **Circumference Calculation**:
- The circumference of the external boundary is given by \( C_{external} = 2\pi r_2 = 2\pi (r_1 + 5) \).
- The circumference of the internal boundary is given by \( C_{internal} = 2\pi r_1 \).

4. **Set Up the Ratio**:
- The ratio of the time taken to travel the external boundary to the internal boundary is given as \( \frac{C_{external}}{C_{internal}} = \frac{20}{19} \).
- Substituting the circumferences into the ratio gives us:
\[ \frac{2\pi (r_1 + 5)}{2\pi r_1} = \frac{20}{19} \]
- The \( 2\pi \) cancels out, simplifying to:
\[ \frac{r_1 + 5}{r_1} = \frac{20}{19} \]

5. **Cross Multiply**:
- Cross multiplying gives:
\[ 19(r_1 + 5) = 20r_1 \]
- Expanding this results in:
\[ 19r_1 + 95 = 20r_1 \]

6. **Rearranging the Equation**:
- Rearranging the equation gives:
\[ 95 = 20r_1 - 19r_1 \]
\[ 95 = r_1 \]

7. **Calculate the Internal Diameter**:
- The internal diameter \( D \) is given by \( D = 2r_1 \):
\[ D = 2 \times 95 = 190 \text{ meters} \]

### Final Answer:
The internal diameter is **190 meters**.

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