April 27th, 2018, 11:48 PM  #1 
Senior Member Joined: May 2008 Posts: 290 Thanks: 76  Typo? Beer soaked speculation follows. A college student emailed me the following right triangle problem: An airplane starts from a station and rises at an angle of 10° with the horizontal. By how many feet will it clear a vertical wall 110 ft. high and 900 ft. from the station? Assume that tan 10° = 0.176. Ans. 48.40 ft. He thinks the given answer may be a typo. I'm somehow inclined to agree. Upon asking where he found this problem, he replied that he got it from Plane and Spherical Trigonometry With Tables by Kells, Kern, Bland (All at the US Naval Academy) 3e 1951 Exercise 6.11 (p. 17) 1951! Awesome. As I see it, the plane is at a station 900 ft. from an imaginary wall 110 ft. high. It then gets a running start of maybe a few hundred ft. then starts to rise at an angle of 10°. If the question "By how many feet will it clear a vertical wall 110 ft. high and 900 ft. from the station?" is simply referring to the base distance the moment the plane rises to the point it reaches a height of 110 ft., then it ought to be just a blind application of the tangent function where tangent 10° = 110/x where it's assumed that tangent 10° = 0.176 We then have x = 625 ft., implying that the running start was 900  625 or 275 ft. Thus my speculation that the given answer may be a typo. Otherwise, my being rusty in basic trigonometry and beer goggles may be at fault. What say you Sir D? 
April 28th, 2018, 12:33 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,099 Thanks: 1093 
the problem may be taking the curve of the spherical earth into consideration

April 28th, 2018, 05:58 AM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,126 Thanks: 914 
No rounding to .176 gives 48.69428..... So 48.4 looks fine to me. 
April 28th, 2018, 07:49 AM  #4 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,126 Thanks: 914 
ANUTter "look": TAN(10) = .176327 : results with 48.69428 then: .176 = 48.69428 / .176327 * .176 = 48.60 So appears that 48.4 is wrong (or a typo). 
April 28th, 2018, 08:42 AM  #5  
Math Team Joined: Jul 2011 From: Texas Posts: 2,767 Thanks: 1422  Quote:
$158.4  110 = 48.4$ what typo?  
April 28th, 2018, 11:10 AM  #6 
Senior Member Joined: May 2008 Posts: 290 Thanks: 76  Beer soaked gratitude follows. So another (probably the intended) interpretation of the problem is that the plane gets a running start before the station (or some kind of line marker) then starts to rise exactly at that point at an angle of 10° and the question "By how many feet will it clear a vertical wall 110 ft. high and 900 ft. from the station?" is simply referring to the distance from the plane's height at a distance of 900 ft. from the station to the top of a wall 110 ft. high. Awesome. Why didn't I think of that? Nothing like a military officer's perspective to sort out a seemingly simple trigonometry problem. Should have worn my brandy goggles. Thanks again skeeter. Last edited by jonah; April 28th, 2018 at 11:14 AM. 

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