Fifty points - it took 4 hours for a biker to travel from one city to another going at a certain speed. on the return trip, the biker traveled at the same speed for the first 100-km and then for the rest of the trip he traveled at a speed which was 10 km/hour slower than the original speed, and thus the return trip took him 30 min longer. find the distance between the two cities.

Either 200 km or 160 km.

Step-by-step explanation:

Distance = speed × time

Let x = the biker's normal speed

Then

(1) d = 4x

For the return trip,

Time = distance/speed

(2)                 4.5 = 100/x + (d -100)/(x - 10)

                    4.5x = 100/x + (4x -100)/(x - 10)     Substituted (1) into (2)

                    4.5x = 100 + x(4x -100)/(x - 10)      Multiplied each side by x

          4.5x(x - 10) = 100(x - 10) + x(4x -100)      Multiplied each side by (x - 10)

          4.5x² - 45x = 100x - 1000 + 4x² -100x   Removed parentheses

          4.5x² - 45x = -1000 + 4x²                       Cancelled terms

          0.5x² - 45x = -1000     Subtracted 4x² from each side

0.5x² - 45x +1000 = 0            Added 1000 to each side

    x² - 90x +2000 = 0           Multiplied each side by 2

    (x - 50)(x - 40) = 0             Factored the quadratic

x - 50 = 0     x - 40 = 0          Applied zero product theorem

      x = 50          x = 40

Substitute in (1)

d = 4 × 50 = 200 km; d = 4  × 40 = 160 km

The distance between the two cities is either 200 km or 160 km.

Check:

   x = 50                                                    x = 40

4.5 = 100/50 + (200 - 100)/(50 - 10)     4.5 = 100/40 + (160 - 100)/(40 - 10)

4.5 = 2  + 100/40                                  4.5 = 2.5 + 60 /30

4.5 = 2 + 2.5                                         4.5 = 2.5 + 2

4.5 = 4.5                                               4.5 = 4.5  

OK

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