Rihanna works in a coffee shop approximately nine miles from her apartment she bikes every day from her apartment to the coffee shop and then back to her apartment in the evening. however, on her way to work, she always stops halfway through to meet her best friend at a park. if the distance from apartment to the park is (5x 2) miles and the trip from the park to the coffee shop is (25x -8) miles long, then what is the value of x?

We will assume that the apartment, the park and the coffee shop and the park all lie on the straight line.

From the givens, we know that:
The total distance from the apartment to the coffee shop = 9 miles.
Distance from the apartment to the park = 5x^2.
Distance from the park to the coffee shop = 25x - 8

The distance from the apartment to the park added to that from the park to the coffee shop will give us the total distance.
This can be interpreted to an equation as follows:
5x^2 + 25x - 8 = 9
5x^2 + 25x - 17 = 0

We will then solve this quadratic equation to get the values of x as follows:
either x = 0.6064 (accepted value)
or x = -5.6064 (rejected value as no distance is in negative)

Distance from the apartment to the park = 5x^2 = 1.84 miles.
Distance from the park to the coffee shop = 25x - 8 = 7.16 miles.