This problem involves finding how many times the front wheel of a tractor will rotate, given the rear wheel's rotations and the difference in their radii.

Step-by-Step Solution:

1. Key Variables:

   • Rear wheel's radius ($r_{\text{rear}}$) = 30 cm
   • Front wheel's radius ($r_{\text{front}}$) = 50 cm
   • Rear wheel's rotations = 60

2. Calculate the Rear Wheel's Circumference: The circumference of a circle is given by $C = 2 \pi r$.

   • For the rear wheel: $C_{\text{rear}} = 2 \pi (30) = 60\pi \, \text{cm}.$

3. Calculate the Total Distance Covered: Distance covered is the number of rotations multiplied by the circumference: $\text{Distance} = \text{Rotations} \times C_{\text{rear}}.$ Substituting: $\text{Distance} = 60 \times 60\pi = 3600\pi \, \text{cm}.$

4. Calculate the Front Wheel's Circumference: Using $C = 2 \pi r$:

   • For the front wheel: $C_{\text{front}} = 2 \pi (50) = 100\pi \, \text{cm}.$

5. Determine Front Wheel Rotations: The total distance covered will be the same for both wheels. Hence, the number of rotations ($n$) for the front wheel is: $n = \frac{\text{Distance}}{C_{\text{front}}}.$ Substituting: $n = \frac{3600\pi}{100\pi} = \frac{3600}{100} = 90.$

Final Answer:

The front wheel will rotate 90 times when the rear wheel rotates 60 times. The correct choice is C) 90.

Would you like further clarification or details? Here are 5 related questions to explore further:

1. How is the relationship between radii and rotations derived in a rolling motion scenario?
2. Can the solution change if the tractor covers uneven terrain?
3. How would the answer change if the rear radius was doubled instead?
4. What is the ratio of rotations if the front and rear radii are interchanged?
5. How does the circumference formula affect other wheel-based motion problems?

Tip: Always verify proportional relationships between distance, rotations, and circumference to ensure consistency in similar problems.