Kartarpur is 64 kilometers from Amritsar.Find to the nearest second the angle subtended at the centre of the earth by the arc joining these two towns, earth being regarded as a sphere of 6400 kilometer radius.

Answer

To solve the problem of finding the angle subtended at the center of the Earth by the arc joining Kartarpur and Amritsar, we can follow these steps:

### Step 1: Understand the relationship between arc length, radius, and angle
The formula relating arc length (L), radius (R), and angle (θ in radians) is given by:
\[ L = R \theta \]

### Step 2: Identify the given values
From the problem, we know:
- Distance between Kartarpur and Amritsar (arc length, L) = 64 km
- Radius of the Earth (R) = 6400 km

### Step 3: Rearrange the formula to solve for θ
We can rearrange the formula to find θ:
\[ \theta = \frac{L}{R} \]

### Step 4: Substitute the values into the equation
Now, substituting the values of L and R into the equation:
\[ \theta = \frac{64 \text{ km}}{6400 \text{ km}} \]

### Step 5: Calculate θ in radians
Calculating the above gives:
\[ \theta = \frac{64}{6400} = \frac{1}{100} \text{ radians} \]

### Step 6: Convert radians to minutes
To convert radians to minutes, we use the conversion factor:
\[ 1 \text{ radian} = \frac{180}{\pi} \times 60 \text{ minutes} \]
Thus,
\[ \theta \text{ in minutes} = \frac{1}{100} \times \frac{180}{\pi} \times 60 \]

### Step 7: Simplify the expression
Calculating this gives:
\[ \theta \text{ in minutes} = \frac{180 \times 60}{100 \pi} = \frac{10800}{100 \pi} = \frac{108}{\pi} \text{ minutes} \]

### Step 8: Calculate the numerical value
Using the approximation \( \pi \approx \frac{22}{7} \):
\[ \theta \text{ in minutes} \approx \frac{108 \times 7}{22} = \frac{756}{22} \approx 34.3636 \text{ minutes} \]

### Step 9: Separate minutes and seconds
The integer part is 34 minutes. To find the seconds, we take the decimal part:
\[ 0.3636 \text{ minutes} \times 60 \text{ seconds/minute} \approx 21.816 \text{ seconds} \]

### Step 10: Round to the nearest second
Rounding 21.816 seconds gives approximately 22 seconds.

### Final Answer
Thus, the angle subtended at the center of the Earth by the arc joining Kartarpur and Amritsar is:
\[ 34 \text{ minutes and } 22 \text{ seconds} \]