Question 924982
If you draw out the triangle, you get 


<img src = "http://i150.photobucket.com/albums/s91/jim_thompson5910/11-18-20142-16-58PM_zps15b89a2e.png">


* <font color="blue">y</font> is the height of the airplane at time t (y is the opposite side)


* <font color="red">350t</font> represents how far along the hypotenuse the plane has traveled


* t is time in seconds


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From the drawing above, we see 


<font color="blue">opposite</font> = <font color="blue">y</font>


<font color="red">hypotenuse</font> = <font color="red">350t</font> 


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So that means we can say



sin(theta) = <font color="blue">opposite</font>/<font color="red">hypotenuse</font>



sin(15) = <font color="blue">y</font>/<font color="red">350t</font>



sin(15) = y/350t



350t*sin(15) = y



(350*sin(15))*t = y



y = (350*sin(15))*t


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The function {{{y = (350*sin(15))*t}}} represents how high the plane is at time t.


y is the vertical distance.
t is time in seconds.


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Plug in y = 25,000 and solve for t



{{{y = (350*sin(15))*t}}}



{{{25000 = (350*sin(15))*t}}}



{{{25000/(350*sin(15)) = t}}}



{{{t = 25000/(350*sin(15))}}}



{{{t = 275.978807511162}}} ... use a <a href="http://web2.0calc.com/">calculator</a> here (type in "<font color="blue">25000/(350*sin(15))</font>" without quotes). 


Special Note: make sure you're in degree mode and NOT in radian mode.



{{{t = 275.98}}} ... round to 2 decimal places



It takes approximately <font size=5 color="red">275.98 seconds</font> for the airplane to climb to 25,000 ft. So that is your answer.



Note: 275.98 seconds = 4 minutes, 35.98 seconds = 4.6 minutes (approximate). This is extra info to possibly think of the timeframe in an easier context.