Answer:
The speed of River's mom drove for 10 miles distance is 0.234 mph
The speed of  River's mom drove for 25 miles is 10.234 mph Â
Step-by-step explanation:
Given as :
The first distance cover by River's mom = 10 miles
The rate of speed = x mph
The second distance cover by river's mom = 25 miles
The rate of speed = x + 10 mph
The total driving time = 45 min
Now, According to question
Time = [tex]\dfrac{\textrm Distance}{\textrm speed}[/tex]
So, 45 = [tex]\dfrac{10}{x}[/tex] + Â [tex]\dfrac{25}{x + 10}[/tex]
Or, 9 = [tex]\dfrac{2}{x}[/tex] + Â [tex]\dfrac{5}{x + 10}[/tex]
Or, 9 × x × ( x + 10 ) = 2 × ( x + 10 ) + 5 x
Or, 9 x² + 90 x = 2 x + 20 + 5 x
Or, 9 x² + 90 x = 7 x + 20
Or, 9 x² + 90 x - 7 x - 20 = 0
Or, 9 x² + 83 x - 20 = 0
Solving this quadratic equation
x = [tex]\frac{-b\pm \sqrt{b^{2}-4\times a\times c}}{2\times a}[/tex]
Or, x = [tex]\frac{-83\pm \sqrt{83^{2}-4\times 9\times (-20)}}{2\times 9}[/tex]
Or, x = [tex]\frac{-83\pm \sqrt{7609}}{18}[/tex]
∴  x = [tex]\frac{-83+87.22}{18}[/tex] , tex]\frac{-83-87.22}{18}[/tex]
I.e x = 0.234 , - 9.4567
We consider x = 0.234
So, The speed for 10 miles distance = x = 0.234 mph
and The speed of 25 miles = x + 10 = 0.234 + 10 = 10.234 mph
Hence The speed of River's mom drove for 10 miles distance is 0.234 mph
and The speed of  River's mom drove for 25 miles is 10.234 mph  Answer
