To determine the correct formula for calculating the fare for [tex]\( x \)[/tex] kilometers traveled by an auto-rickshaw, we can break down the problem step by step.
1. Initial Cost for the First Kilometer:
- The fare for the first [tex]\( 1 \)[/tex] kilometer (km) is fixed at Rs. [tex]\( 10 \)[/tex].
2. Cost for Additional Distance:
- For every kilometer beyond the first kilometer, the fare is Rs. [tex]\( 4.50 \)[/tex] per kilometer.
Based on these rules, let's consider each answer choice and check if it aligns with the given fare structure:
A. [tex]\( 4.5(x-1) + 10 \)[/tex]:
This formula suggests:
- After the first kilometer, only the additional kilometers are charged at the rate of Rs. [tex]\( 4.50 \)[/tex] per km.
- So, for [tex]\( x \)[/tex] kilometers, if [tex]\( x \leq 1 \)[/tex], the fare is Rs. [tex]\( 10 \)[/tex].
- If [tex]\( x > 1 \)[/tex], the fare would be Rs. [tex]\( 10 \)[/tex] (first km) [tex]\( + 4.5(x-1) \)[/tex] (remaining kilometers).
This seems correct because it correctly adds the base charge and charges the remaining distance separately.
B. [tex]\( 10(x-1) + 4.5 \)[/tex]:
This formula suggests:
- For [tex]\( x - 1 \)[/tex] kilometers, the fare is Rs. [tex]\( 10 \)[/tex] per kilometer.
- Adding Rs. [tex]\( 4.5 \)[/tex] as a flat fee.
This formula incorrectly applies the Rs. [tex]\( 10 \)[/tex] rate to the kilometers beyond the first km, and the additional Rs. [tex]\( 4.5 \)[/tex] doesn't fit the given structure.
C. [tex]\( 4.5x + 10 \)[/tex]:
This formula suggests:
- Rs. [tex]\( 4.5 \)[/tex] per km for [tex]\( x \)[/tex] kilometers.
- Adding Rs. [tex]\( 10 \)[/tex] as an additional flat fee.
This formula does not correctly account for the specific rate for the first km being different from the subsequent kilometers.
D. [tex]\( 10x + 4.5 \)[/tex]:
This formula suggests:
- A flat rate of Rs. [tex]\( 10 \)[/tex] per kilometer for [tex]\( x \)[/tex] kilometers.
- Adding Rs. [tex]\( 4.5 \)[/tex] as an additional fee.
This formula also does not correctly account for the different rates for the first kilometer and the subsequent kilometers.
Conclusion:
The correct general rule to calculate the fare for [tex]\( x \)[/tex] kilometers traveled is:
A. [tex]\( 4.5(x-1) + 10 \)[/tex]
