The radius of the wheel of a vehicle is 42 cm. How many revolutioins will it complete in a 19.8-km-long journey?

Explanation

To solve the problem of how many revolutions a wheel with a radius of 42 cm will complete in a 19.8 km journey, we can follow these steps:

### Step-by-Step Solution:

1. **Convert the distance from kilometers to centimeters:**
\[ \text{Distance} = 19.8 \text{ km} = 19.8 \times 1000 \text{ m} = 19800 \text{ m} = 19800 \times 100 \text{ cm} = 1980000 \text{ cm} \]

**Hint:** Remember that 1 km = 1000 m and 1 m = 100 cm.

2. **Calculate the circumference of the wheel:**
The circumference \( C \) of a circle (which represents the distance covered in one complete revolution of the wheel) is given by the formula:
\[ C = 2 \pi R \]
where \( R \) is the radius of the wheel. Substituting \( R = 42 \) cm:
\[ C = 2 \times \frac{22}{7} \times 42 \]

**Hint:** Use \( \pi \approx \frac{22}{7} \) for calculations.

3. **Calculate the circumference:**
\[ C = 2 \times \frac{22}{7} \times 42 = \frac{1848}{7} \text{ cm} \]
\[ C \approx 264 \text{ cm} \]

**Hint:** Perform the multiplication and division carefully to find the circumference.

4. **Determine the number of revolutions:**
The number of revolutions \( N \) can be calculated by dividing the total distance traveled by the circumference of the wheel:
\[ N = \frac{\text{Total Distance}}{\text{Circumference}} = \frac{1980000 \text{ cm}}{264 \text{ cm}} \]

**Hint:** Ensure that both the total distance and circumference are in the same units (cm).

5. **Calculate the number of revolutions:**
\[ N = \frac{1980000}{264} \approx 7500 \]

**Hint:** Perform the division to find the total number of revolutions.

### Final Answer:
The wheel will complete approximately **7500 revolutions** during the 19.8 km journey.