# passenger plane flew from Texas to Frankfurt and back: The round trip is 9900 km. The return trip took hours less than the trip to Frankfurt because of the 50 km/hr wind over the ocean on the return trip. Assume there Was no wind On the trip to Frankfurt. Let x km/h represent the average speed of the plane in still air:

a). Express the trip time from Texas to Frankfurt as a rational function of variable
b) . Express the trip time from Frankfurt to Texas as a rational function of variable x
c) Write an equation representing the relationships between the two trip times
d)Find the average speed of the plane in still air [3marksl e). How long did the round trip take?

## speed of plane in still air --- x km/h <--- speed for T to F

speed of plane on F to T trip ---- x + 50 km/h

time taken to go from Texas to Frankfurt = 4950/x hours
time taken to go from Frankfurt to Texas = 4950/(x+50)

"The return trip took hours less than the trip to Frankfurt"
------ missing the number of hours, let's assume the difference was 3 hours, then the equation would be

4950/x - 4950/(x+50) = 3

multiply each term by x(x-50)
4950x- 4950(x-50) = 3x(x-2)

Once you replace 3 with the correct difference in time, solve for x
using the quadratic formula. Make sure to use only the positive answer
since x is the speed of the plane.

sub in your x into the two "times" expressions above, then add them up
for a total time

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## a) The trip time from Texas to Frankfurt can be expressed as a rational function of the variable x as follows:

Trip Time (Texas to Frankfurt) = Distance/Speed = (9900)/(x + 50)

b) The trip time from Frankfurt to Texas can be expressed as a rational function of the variable x as follows:
Trip Time (Frankfurt to Texas) = Distance/Speed = (9900)/(x - 50)

c) The equation representing the relationship between the two trip times can be written as:
Trip Time (Texas to Frankfurt) = Trip Time (Frankfurt to Texas)

d) To find the average speed of the plane in still air, we can set the two trip times equal to each other and solve for x:

(9900)/(x + 50) = (9900)/(x - 50)

Cross multiplying, we get:
(9900)(x - 50) = (9900)(x + 50)

Expanding both sides, we get:
9900x - 495000 = 9900x + 495000

Simplifying, we find:
-495000 = 495000

This is a contradiction, which means there is no valid solution for x. Therefore, there is no average speed in still air that satisfies the given conditions.

e) Since there is no valid solution for the average speed of the plane in still air, we cannot determine the exact round trip time.

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## a) To express the trip time from Texas to Frankfurt as a rational function of the variable x, we need to consider the distance and the speed of the plane. The distance from Texas to Frankfurt is given as 9900 km. Since there is no wind on the trip to Frankfurt, we can assume the average speed of the plane in still air as x km/h.

Using the formula: time = distance / speed, the trip time from Texas to Frankfurt can be expressed as:

Time_TF = 9900 / x

b) To express the trip time from Frankfurt to Texas as a rational function of the variable x, we need to consider the distance and the speed of the plane. The same distance of 9900 km needs to be covered on the return trip, but this time there is a 50 km/hr wind over the ocean, which affects the speed of the plane.

Since the plane speed is affected by the wind, we need to subtract the wind speed from the plane's actual speed to get its effective speed. So the effective speed of the plane on the return trip is (x - 50) km/h.

Using the formula: time = distance / speed, the trip time from Frankfurt to Texas can be expressed as:

Time_FT = 9900 / (x - 50)

c) The relationship between the two trip times is that the return trip took hours less than the trip to Frankfurt because of the 50 km/hr wind. This can be written as:

Time_TF = Time_FT + hours less

d) To find the average speed of the plane in still air (x), we need to solve the equation derived from the relationship between the two trip times. From part c), we have:

Time_TF = Time_FT + hours less

9900 / x = 9900 / (x - 50) + hours less

To solve for x, we need to know the specific value of "hours less" or have additional information.

e) To find the total time for the round trip, we can add the trip time from Texas to Frankfurt (Time_TF) and the trip time from Frankfurt to Texas (Time_FT).

Total time = Time_TF + Time_FT = 9900 / x + 9900 / (x - 50)