Question 88835
{{{d=rt}}} Start with the blank rate equation



{{{720=(x+30)*t[1]}}} Now plug in the given values. This is the rate equation <b>with</b> the wind.


{{{720/(x+30)=t[1]}}} Solve for t




{{{720=(x-30)*t[2]}}} Now plug in the given values. This is the rate equation <b>against</b> the wind.


{{{720/(x-30)=t[2]}}} Solve for t


Now combine the two "t" equations and set them equal to 10


{{{t[1]+t[2]=10}}}


{{{720/(x+30)+720/(x-30)=10}}} Plug in {{{t[1]=720/(x+30)}}} and {{{t[2]=720/(x-30)}}}



{{{(x+30)(x-30)(720/(x+30)+720/(x-30))=10(x+30)(x-30)}}} Multiply both sides by {{{(x+30)(x-30)}}}



{{{720(x-30)+720(x+30)=10(x+30)(x-30)}}} Distribute the left side



{{{720(x-30)+720(x+30)=10(x^2-900)}}} Foil the left side


{{{720(x-30)+720(x+30)=10x^2-9000}}} Multiply the right side


{{{720x-21600+720(x+30)=10x^2-9000}}} Distribute and multiply {{{720(x-30)}}}


{{{720x-21600+720x+21600=10x^2-9000}}} Distribute and multiply {{{720(x+30)}}}


{{{1440x=10x^2-9000}}} Combine like terms on the left side


{{{0=10x^2-1440x-9000}}} Subtract 1440x from both sides



Now let's use the quadratic formula to solve for x:



Starting with the general quadratic


{{{ax^2+bx+c=0}}}


the general solution using the quadratic equation is:


{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}


So lets solve {{{10*x^2-1440*x-9000=0}}} ( notice {{{a=10}}}, {{{b=-1440}}}, and {{{c=-9000}}})


{{{x = (--1440 +- sqrt( (-1440)^2-4*10*-9000 ))/(2*10)}}} Plug in a=10, b=-1440, and c=-9000




{{{x = (1440 +- sqrt( (-1440)^2-4*10*-9000 ))/(2*10)}}} Negate -1440 to get 1440




{{{x = (1440 +- sqrt( 2073600-4*10*-9000 ))/(2*10)}}} Square -1440 to get 2073600




{{{x = (1440 +- sqrt( 2073600+360000 ))/(2*10)}}} Multiply {{{-4*-9000*10}}} to get {{{360000}}}




{{{x = (1440 +- sqrt( 2433600 ))/(2*10)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (1440 +- 1560)/(2*10)}}} Simplify the square root




{{{x = (1440 +- 1560)/20}}} Multiply 2 and 10 to get 20


So now the expression breaks down into two parts


{{{x = (1440 + 1560)/20}}} or {{{x = (1440 - 1560)/20}}}


Lets look at the first part:


{{{x=(1440 + 1560)/20}}}


{{{x=3000/20}}} Add the terms in the numerator

{{{x=150}}} Divide


So one answer is

{{{x=150}}}




Now lets look at the second part:


{{{x=(1440 - 1560)/20}}}


{{{x=-120/20}}} Subtract the terms in the numerator

{{{x=-6}}} Divide


So another answer is

{{{x=-6}}}


So our solutions are:

{{{x=150}}} or {{{x=-6}}}





Since a negative speed doesn't make any sense, our solution is {{{x=150}}}


So the plane's speed in still air is 150 mph