The direction of movement of both are perpendicular to each other
Let us say Alice's speed = x mph
then speed of Lisa = x- 9 mph
The distance traveled by Lisa in 1 hour = 1 times ( x-9 ) = x-9 miles
the distance traveled by Alice in 1 hour = 1 times x = x miles
the distance between them is : the length of the hypotaneous of the triangle with perpendicular sides being x and x-9
That is : ( x^2 + ( x-9)^2 ) ^(1/2) which is given to be equal to 65
x^2 + ( x-9)^2 = ( 65)^2
x^2 + x^2 + 9^2 - 2* 9 x = 4225
2x ^2 - 18x = 4225 - 9^2
2x^2 - 18x = 4144
dividing each term by 2
x^2 - 9x = 2072
x^2 - 9x - 2072 = 0
we plug a= 1 b= -9 c=-2072 in the quadratic formula
x= [-b +/- ( b^2 - 4ac ) ]/ 2a
x= [ 9 +/- ( (-9)^2 - 4 ( 1)( -2072) ] / 2* 1
x= 50.24 miles /hour
Answer : Alice's average speed is 50.24 miles.
