Question 872087
The current river flows at the rate of 4mph.
 In order for a boat to travel 24 miles up river and then return in a total of 5 hours, how fast must the boat be able to travel in still water?
:
Let s = speed in still water
Then
(s-4) = effective speed upstream
and
(s+4) = effective speed downstream
:
Write a time equation, time = dist/speed
:
up-str time + down-str time = 5 hrs
{{{24/((s-4))}}} + {{{24/((s+4))}}} = 5
Multiply by (s-4)(s+4)
(s-4)(s+4)*{{{24/((s-4))}}} + (s-4)(s+4){{{24/((s+4))}}} = 5(s-4)(s+4)
Cancel the denominators, you have
24(s+4) + 24(s-4) = 5(s^2-16)
24s + 96 + 24s - 96 = 5s^2 - 80
48s = 5s^2 - 80
Combine to form a quadratic equation on the right
0 = 5s^2 - 48s - 80
This will not factor, use the quadratic formula
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}, where
x = s; a=5; b=-48; c=-80
{{{x = (-(-48) +- sqrt( -48^2-4*5*-80 ))/(2*5) }}}
I'll let you do the math here, I got a positive solution of 11.05 mph
:
:
Check this in the original equation