Question 389640
Let x = number of seconds that the sleigh and airplane have been traveling


In this case, a picture is worth a thousand words (and symbols). So let's draw one



<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/santassleighvsairplane.png">



In the picture above, we can see that Santa has traveled {{{5450x}}} feet (first segment shown in red). Recall that {{{d=rt}}} (distance = rate*time). In this case, the rate is 5450ft/sec, so r=5450 and he has been traveling for 'x' seconds. So t=x



Likewise, the airplane has been traveling at a speed of 1240 ft/sec for 'x' seconds as well. So it has traveled {{{1240x}}} ft (last segment shown in red).



The final middle segment has a length of 10,000 ft. because we <u>want</u> the distance between the two to be 10,000 ft at t = x seconds.



Since the entire distance between them is 25,000 ft, this means that the length of the entire red line is 25,000 ft. So the individual pieces must add to 25000.



So this means that {{{5450x+10000+1240x=25000}}}. 



Let's solve for 'x' to find the time it will take for Rudolf to spot the airplane.




{{{5450x+10000+1240x=25000}}} Start with the given equation.



{{{6690x+10000=25000}}} Combine like terms on the left side.



{{{6690x=25000-10000}}} Subtract {{{10000}}} from both sides.



{{{6690x=15000}}} Combine like terms on the right side.



{{{x=(15000)/(6690)}}} Divide both sides by {{{6690}}} to isolate {{{x}}}.



{{{x=500/223}}} Reduce.



{{{x=2.24215246636771}}} Use a calculator to find the decimal approximation



So at approximately {{{2.24215246636771}}} seconds, the sleigh and the airplane will be 10,000 ft apart.



So it will take about {{{2.24215246636771}}} seconds for Rudolf to spot the airplane.



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b) I'll let you try this part on your own. I recommend using the picture above as a guide. 



Hint: Since the sleigh needs 2500 ft of clearance, the distance between the plane and the sleigh has been reduced to 2500 ft (instead of 10,000). Once you find this new value of t, subtract it from the answer in part a) to find the elapsed time window that Rudolf has (ie the window for his reaction). If you're still stuck, let me know.



If you need more help, email me at <a href="mailto:jim_thompson5910@hotmail.com?Subject=Algebra%20Help">jim_thompson5910@hotmail.com</a>


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Jim