Alina spent no more than $45 on gas for a road trip. the first gas station she used charged $3.50 per gallon and the second gas station charged $4.00 per gallon.
Which inequality relates the number of gallons of gas she bought at the first station, x, the number of gallons of gas she bought at the second station, y,
and the total amount she paid? what are the possible values of y?
A. 3.5x 4y = 45, y = 11.25
B. 3.5x 4y = 45, 0 = y = 10.375
C. 3.5x 4y = 45, 0 = y = 11.25
D. 3.5x 4y = 45, y = 10.375
The Correct Answer and Explanation is :
To find the correct inequality and possible values of ( y ), we first need to represent the total cost of gas purchased from two different gas stations.
Let:
- ( x ) = the number of gallons of gas bought at the first station (which charges $3.50 per gallon)
- ( y ) = the number of gallons of gas bought at the second station (which charges $4.00 per gallon)
The total cost ( C ) can be expressed as:
[
C = 3.50x + 4.00y
]
Since Alina spent no more than $45 on gas, we can write the inequality as:
[
3.50x + 4.00y \leq 45
]
Now, to analyze the possible values of ( y ), we will isolate ( y ) in the inequality:
- Rearranging the inequality gives us:
[
4.00y \leq 45 – 3.50x
]
[
y \leq \frac{45 – 3.50x}{4}
]
Next, we need to determine the maximum value of ( y ) when ( x = 0 ) (i.e., when all the gas is purchased from the second station):
[
y \leq \frac{45 – 3.50(0)}{4} = \frac{45}{4} = 11.25
]
This indicates that ( y ) can take any value from ( 0 ) to ( 11.25 ), provided that the total cost does not exceed $45.
Now, let’s summarize the options provided:
- A. ( 3.5x + 4y = 45, y = 11.25 )
- B. ( 3.5x + 4y = 45, 0 \leq y \leq 10.375 )
- C. ( 3.5x + 4y = 45, 0 \leq y \leq 11.25 )
- D. ( 3.5x + 4y = 45, y = 10.375 )
Given our analysis, the correct option is:
C. ( 3.5x + 4y = 45, 0 \leq y \leq 11.25 )
This option correctly reflects the total cost equation and the range of values for ( y ), which indicates that Alina can purchase up to 11.25 gallons of gas from the second station without exceeding her budget. Thus, as long as ( y ) remains within this interval, the inequality holds true, allowing Alina to manage her expenses efficiently on her road trip.