A car traveled 243 miles on a trip. During 63 miles of the trip, the car traveled through the city and had gas mileage that was 9 fewer miles per gallon that the gas mileage on the highway. If the total amount of gasoline used for the trip was 9 gallons, find the gas mileage of the car on the highway. Please help ASAP!!! :(
gas used is miles / mpg, so add up the gas for each leg of the trip:
63/(x-9) + 180/x = 9
Now just solve for x. Discard the meaningless answer.
To find the gas mileage of the car on the highway, we need to determine the number of miles the car traveled on the highway and the corresponding gas mileage during that portion of the trip.
Let's denote the gas mileage on the highway as x miles per gallon.
We know that during the 63 miles through the city, the car had a gas mileage that was 9 fewer miles per gallon than on the highway. Therefore, the gas mileage in the city is (x - 9) miles per gallon.
We can set up an equation to represent the total amount of gasoline used for the trip:
Total gas used = Gas used on the highway + Gas used in the city
Given that the total amount of gasoline used for the trip was 9 gallons, we have:
9 = (highway distance / x) + (city distance / (x - 9))
We are given that the car traveled a total distance of 243 miles, with 63 miles through the city. Therefore, the distance traveled on the highway is 243 - 63 = 180 miles.
Substituting these values into the equation, we have:
9 = (180 / x) + (63 / (x - 9))
Now, we can solve this equation to find the gas mileage on the highway.
To do this, let's first clear the fractions by multiplying through by the common denominator, which is x(x-9):
9x(x-9) = 180(x - 9) + 63x
Expanding and simplifying:
9x^2 - 81x = 180x - 1620 + 63x
9x^2 - 81x - 180x + 63x - 1620 = 0
9x^2 - 198x - 1620 = 0
Next, we can factor or use the quadratic formula to solve this equation. Using the quadratic formula is the most straightforward approach:
x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 9, b = -198, and c = -1620. Substituting these values into the formula:
x = (-(-198) ± √((-198)^2 - 4(9)(-1620))) / (2(9))
Simplifying:
x = (198 ± √(39204 + 58320)) / 18
x = (198 ± √(97524)) / 18
Now, we calculate the two possible values for x:
x = (198 ± 312.5) / 18
The two possible solutions are:
1) x = (198 + 312.5) / 18 = 510.5 / 18 ≈ 28.36 miles per gallon
2) x = (198 - 312.5) / 18 = -114.5 / 18 ≈ -6.38 miles per gallon
Since gas mileage cannot be negative, we discard the second solution.
Therefore, the gas mileage of the car on the highway is approximately 28.36 miles per gallon.