Question 152709
The speed of an airplane in still air is 250 km. The plane travels 632 km
against the wind and 1343 km with the wind in a total time of 15 hours. What is the speed of the wind? 
:
Write down what you know:
Let x = speed of the wind
then
(250+x) = the speed of the plane with the wind
and
(250-x) = the speed against the wind
:
Write a time equation: Time = Dist/speed
:
Time with + time against = 15 hrs
{{{1343/((250+x))}}} + {{{632/((250-x))}}} = 15
;
Multiply equation by the common denominator: (250+x)(250-x), results
1343(250-x) + 632(250+x) = 15(250+x)(250-x)
:
335750 - 1343x + 158000 + 632x = 15(62500 - x^2); FOILed
:
493750 - 711x = 937500 - 15x^2
Arrange as a quadratic equation
15x^2 - 711x + 493750 - 937500 = 0
:
15x^2 - 711x - 443750 = 0
:
A nasty equation to be solved by the quadratic formula
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
In this problem a=15; b=-711; c= -443750
{{{x = (-(-711) +- sqrt(-711^2 - 4 * 15 *-443750 ))/(2*15) }}}
:
{{{x = (711 +- sqrt(505521 + 26625000 ))/(30) }}}
:
{{{x = (711 +- sqrt(27130521 ))/(30) }}}
The positive solution
{{{x = (711 + 5208.7)/(30) }}}
:
{{{x = 5919.7/30 }}}
x = 197.3 is the speed of the wind
:
:
Check this rather extreme solution by finding the total time:
Speed with = 447.3; against = 52.7
Using a calc
1343/447.3 + 632/52.7 =
3.00 + 11.99 = 14.99 ~ 15 hrs