SOLUTION: A plane made a two-way flight from Miami to Dublin. The trip from Miami to Dublin took four more hours than the return trip from Dublin to Miami. The cause for the difference in

Algebra ->  Functions -> SOLUTION: A plane made a two-way flight from Miami to Dublin. The trip from Miami to Dublin took four more hours than the return trip from Dublin to Miami. The cause for the difference in       Log On


   

Question 1204831: A plane made a two-way flight from Miami to Dublin. The trip from Miami to Dublin took four more hours than the return trip from Dublin to Miami. The cause for the difference in flight times were winds over the ocean, which averaged 50 km/hr on the return trip. Assume there was no wind on the trip to Dublin. The total distance travelled both ways was 9900 km. Let x represent the average speed of the plane in still air (no wind).
a) Represent time going from Dublin to Miami as a rational expression.
b) Represent the time going from Miami to Dublin as a rational expression.
c) Write an equation representing the relationships between the times.
d) What was the average rate of the plane in still air.
e) How long did the round trip take?
Found 3 solutions by josgarithmetic, mananth, greenestamps:
Answer by josgarithmetic(39621) About Me  (Show Source):
You can put this solution on YOUR website!
                 SPEED      TIME      DISTANCE
GOING TO           r        x+4       9900/2=4950
COMEING BACK       r+50     x         9900/2=4950

Note that the variables here were assigned differently than your exercise instructed to do. x here is time for the return trip and r is for speed of the plane absent of any wind. Notice, no wind involved for the return trip.
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system%28rx%2B4r=4950%2Crx%2B50x=4950%29

4950-4r=4950-50x
4r=50x
2r=25x
r=25x%2F2

substiture r in first equation.
%2825x%2F2%29x%2B4%2825x%2F2%29=4950
%2825%2F2%29x%5E2%2B%284%2A25%2F2%29x-4950=0
highlight_green%28x%5E2%2B4x-396=0%29

x=%28-4%2B-+sqrt%2816%2B4%2A396%29%29%2F2 must use the positive s.r.
x=%28-4%2Bsqrt%2816%2B4%2A396%29%29%2F2
x=%28-4%2Bsqrt%281600%29%29%2F2
x=%28-4%2B40%29%2F2=-2%2B20=22--------highlight%28x=22%29------Time hours for the return trip.
Therefore r=25x%2F2
r=25%2A22%2F2=25%2A11
highlight%28r=275%29---------speed of plane if no wind

TIME for total round trip, 22%2B4%2B22=highlight%2848%2Ahours%29

Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
A plane made a two-way flight from Miami to Dublin. The trip from Miami to Dublin took four more hours than the return trip from Dublin to Miami. The cause for the difference in flight times were winds over the ocean, which averaged 50 km/hr on the return trip. Assume there was no wind on the trip to Dublin. The total distance travelled both ways was 9900 km. Let x represent the average speed of the plane in still air (no wind).
a) Represent time going from Dublin to Miami as a rational expression.
b) Represent the time going from Miami to Dublin as a rational expression.
c) Write an equation representing the relationships between the times.
d) What was the average rate of the plane in still air.
e) How long did the round trip take?
Let x represent the average speed of the plane in still air
One way distance = 4950
Time taken = 4950/x
From Dublin to Miami there was a tail wind since it took less time
So speed on return trip = 4950/(x+50)
Time Miami to Dublin -Time Dublin to miami =4 hours
4950/x - 4950/(x+50)= 4
multiply equation by x(x+50)
4950(x+50)-4950x = 4x(x+50)
4950x +247500-4950x = 4x^2+200x

4x^2+200x-247500=0
divide by 4
x^2+50x-61875=0
Solve the quadratic equation by any method
x= 225
Miami to Dublin 225 mph
Return trip speed = 225+50 = 175 mph
Time to Dublin + Time to Miami = total time
(4950/225) + (4950/275)

22 hours + 18 hours =40 hours
Now you can answer the questions yourself

Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


a) time = distance/speed: 4950%2Fx

b) time = distance/speed: 4950%2F%28x-50%29

c) time return trip at lower speed was 4 hours longer: 4950%2F%28x-50%29=4950%2Fx%2B4

d) Solve the equation....

4950%2F%28x-50%29=4950%2Fx%2B4

Clear fractions by multiplying by %28x%29%28x-50%29

4950x=4950%28x-50%29%2B4%28x%5E2-50x%29
4x%5E2-200x-247500=0
x%5E2-50x-61875=0

Trying to solve that equation by factoring will take a long time; possibly the best way to finish the problem is to use a graphing calculator or some other tool.

Or here is an algebraic technique that you can use to work this kind of problem.

We are looking for two numbers whose difference is 50 and whose product is 61875.

Let the two numbers that differ by 50 be y+25 and y-25. Then

%28y%2B25%29%28y-25%29=61875
y%5E2-625=61875
y%5E2=62500
y=250

The two speeds are y+25 = 275 km/h and y-25 = 225 km/h.

ANSWER: the speed of the plane in still air is 275 km/h.

e) 4950/275 = 18; 4950/225 = 22; total time 18+22 = 40 hours.