Rosea drives her car 30 kilometers to the train station, where she boards a train to complete her trip. The total trip is 120 kilometers. The average speed of the train is 20 kilometers per hour faster than that of the car. At what speed must she drive her car if the total time for the trip is less than 2.5 hours?
distance = velocity x time or
time = distance / velocity
total time=d1/v1+d2/v2=2.5 hrs (1)
where d1 = car distance = 30 (2)
v1 = car speed
d2 = train distance = 120 - d1 (3)
v2 = train velocity = v1+20 (4)
combine (1), (2), (3), and (4) and solve for v1
300 and 200000
idk can you show me how please
To solve this problem, we can use the concept of time-speed-distance relationship.
Let's assume that Rosea drives her car at a speed of x km/h.
Given that Rosea drives her car for 30 kilometers to the train station, the time taken for this part of the trip can be calculated using the formula: time = distance / speed. Therefore, the time taken for the car journey is 30 / x hours.
Now, since the total trip is 120 kilometers, and she drove 30 kilometers, the remaining distance covered by the train is 120 - 30 = 90 kilometers.
Let's assume the speed of the train is x + 20 km/h (as given it is 20 km/h faster than the car's speed). Using the same formula, the time taken for the train journey is 90 / (x + 20) hours.
We are given that the total time for the trip is less than 2.5 hours. Therefore, the sum of the car journey time and train journey time should be less than 2.5 hours:
30 / x + 90 / (x + 20) < 2.5
To solve this inequality, we can multiply through by x(x + 20) to eliminate the denominators:
30(x + 20) + 90x < 2.5x(x + 20)
Now, simplify and rearrange the equation:
30x + 600 + 90x < 2.5x^2 + 50x
Combine like terms:
120x + 600 < 2.5x^2 + 50x
Rearrange to form a quadratic equation:
0 < 2.5x^2 - 70x + 600
Now, we can solve this quadratic equation to find the range of acceptable speeds for the car.