Question 264834: A single automobile is used to make two trips. The first trip took 5 hours and 39 minutes to complete, and consumed 10.11 gallons of fuel. During this trip, the vehicle travelled 215 miles at “highway speeds” and 78 miles at “city speeds.” The second trip took 7 hours and 37 minutes to complete, and consumed 12.5 gallons of fuel. During this trip, the vehicle traveled 183 miles at “highway speeds” and 156 miles at “city speeds.”
1. What is the overall, average fuel efficiency (MPG) for each of these trips? 2. What is the overall, average speed (MPH) for each of these trips? 3. What is the average fuel efficiency at “highway speeds?” 4. What is the average speed at “highway speeds?”
5. What is the average fuel efficiency at “city speeds?” 6. What is the average speed at “city speeds?”
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! first trip took 5 hours and 39 minutes.
fuel used was 10.11 gallons.
215 miles at highway speeds.
78 miles at city speeds.
second trip took 7 hours and 37 minutes.
fuel used was 12.5 gallons.
183 miles at highway speeds.
156 miles at city speeds.
first trip:
hours were 5 + 39/60 = 5.65
total miles were 215 + 78 = 293
gallons of fuel were 10.11
average speed overall was 293/5.65 = 51.85840708 miles per hour.
average fuel mileage was 293/10.11 = 28.98120673 miles per gallon.
second trip:
hours were 7 + 37/60 = 7.6166666666
total miles were 183 + 156 = 339
gallons of fuel were 12.5
average speed overall was 339/7.6166666666 = 44.50765864 miles per hour.
average fuel mileage was 339/12.5 = 27.12 miles per gallon.
if you assume that highway speeds were the same for both trips and you assume that city speeds were the same for both trips, then you should be able to set up simultaneous equations to solve for city and highway speeds.
if we are talking the average highway speeds for both trips and the average city speeds for both trips, then the assumption that they are the same for both trips is valid.
basic formula is rate * time = distance.
related formulas are:
time = distance / rate.
rate = distance / time.
since we know the total time, and we are assuming that the highway rates and the city rates are the same for both trips, then the formula that we will use is time = distance / rate.
if we let x = rate of travel on the highway and we let y = rate of travel in the city, then:
highway miles / x + city miles / y = total time
the equations we need to solve are:
first equation (trip 1): 215/x + 78/y = 5.65
second equation (trip 2): 183/x + 156/y = 7.6166666666
we multiply the first equation by 2 to get:
first equation (trip 1): 430/x + 156/y = 11.3
second equation (trip 2): 183/x + 156/y = 7.6166666666
we subtract the second equation from the first to get:
247/x = 3.6833333333
we solve for x to get:
x = 67.05882354
we solve for y to get:
y = 31.91672649
we can do the same for city mileage and highway mileage as long as we assume that the city mileage and the highway mileage for both trips was the same.
since we are taking about the average city mileage and highway mileage for both trips, then this assumption is valid.
the formula for mileage is number of miles per gallon equals number of miles divided by number of gallons.
we get:
miles per gallon = distance / gallons
related formula is gallons = distance / miles per gallon.
since we know the overall gallons, and we are assuming that the number of miles per gallon for highway and city are the same for both trips, then we'll use the related formula.
we let x = miles per gallon highway
we let y = miles per gallon city
our formulas are:
first equation (trip 1): 215/x + 78/y = 10.11
second equation (trip 2): 183/x + 156/y = 12.5
we multiply the first equation by 2 to get:
first equation (trip 1): 430/x + 156/y = 20.22
second equation (trip 2): 183/x + 156/y = 12.5
we subtract second equation from first equation to get:
247/x = 7.72
we solve for x to get:
x = 31.99481865 miles per gallon.
we solve for y to get:
y = 23.00775046 miles per gallon.
we are done.
confirmation by plugging into the original equations is necessary.
I did that and the answers are confirmed to be correct based on the assumptions made.
those assumptions were:
highway speed and city speed were the same for both trips.
highway miles per gallon and city miles per gallon were the same for both trips.
this allowed those values to be solved using simultaneous equations.
the questions to be answered were:
1. What is the overall, average fuel efficiency (MPG) for each of these trips?
average fuel efficiency for trip 1 was 28.98120673 miles per gallon.
average fuel efficiency for trip 2 was 27.12 miles per gallon.
2. What is the overall, average speed (MPH) for each of these trips?
average speed for trip 1 was 51.85840708 miles per hour.
average speed for trip 2 was 44.50765864 miles per hour.
3. What is the average fuel efficiency at “highway speeds?”
Average fuel efficiency at highway speeds was 31.99481865 miles per gallon.
4. What is the average speed at “highway speeds?”
Average speed at highway speeds was 67.05882354 miles per hour.
5. What is the average fuel efficiency at “city speeds?”
Average fuel efficiency at city speeds was 23.00775046 miles per gallon.
6. What is the average speed at “city speeds?”
Average speed at city speeds was 31.91672649 miles per hour.
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