Determing the constant speed of a motorboat

Question

A motorboat heads upstream a distance of 24 miles on a river with current running 3 miles per hour. The trip up and back takes 6 hours. Assuming the motorboat maintained a constant speed relative to the water, what was its speed?

Answer 1

Let v = the speed of the boat, t1 the time of the up stream trip (24 miles) and t2 the time of the downstream trip (24 miles). Then (v+3)t1 = 24., and (v-3)t2 = 24 and t1=24/(v+3) and t2 = 24/(v-3). We are given that t1+t2=6(hrs).

From this the resulting quadratic equation is v^2 - 8v -9 = 0 with solution v=9, v=-1. The latter term can be ruled out and we have the speed of the boat v = 9(mi/hr) or 2hrs upstream and 4hrs downstream.

Answer 2

Let x = speed of boat without current

 

Speed with the current = x + 3

Speed against current = x - 3

 

Distance = (Rate)(Time)   So, Time = (Distance)/(Rate)

 

Time upstream = 24/(x-3)

Time downstream = 24/(x+3)

 

Total time = 6 hours

 

So, 24/(x-3) + 24/(x+3) = 6

 

Multiply through by the LCD, (x-3)(x+3):

 

24(x+3) + 24(x-3) = 6(x-3)(x+3)

 

24x + 72 + 24x - 72 = 6(x² - 9)

 

48x  = 6x² - 54

 

6x² - 48x - 54 = 0

 

x² - 8x - 9 = 0

 

(x - 9)(x + 1) = 0

 

x = 9 or x = -1

 

Since x can't be negative, x = 9 mph