Answer
= 4.8 miles per hours.
Step-by-step explanation
Let the time taken to run down the river to be t hours.
The time the bout takes to run up the river will be (t+1.5) hours.
Let the rate of the boat to be x miles per hour.
So the speed of the boat up the river will be (x-2) miles per hour due to the drag the the current.
The speed of the boat down the river will be (x+2).
Distance = 12 miles
With this information we can form two equations and solve them simultaneously.
distance = speed × time
12 = (t+3/2)(x-2)
12 = tx-2t+1.5x-3
tx - 2t + 1.5x =15------------(i)
12 = t(x+2)
t = 12/(x+2)------------------(ii)
Substitute t in equation (i) with 12/(x+2)
tx - 2t + 1.5x =15
12x/(x+2) - 2x/(x+2) +1.5x = 15
12x - 2x +1.5x(x+2) = 15(x+2)
10x + 1.5x²+3x=15x+30
1.5x²-2x-30 = 0
3x² - 2x - 60 = 0
Using quadratic formula;
x = (--2 (+ or -)√(2²+4×3×60))÷(2×3)
x = 4.8 or
x= -4.15.
Since speed cannot be negative, x = 4.8.
Rate of the boat = 4.8 miles per hours.