Typo?

May 2008
302
81
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?
 

romsek

Math Team
Sep 2015
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USA
the problem may be taking the curve of the spherical earth into consideration
 

Denis

Math Team
Oct 2011
14,592
1,026
Ottawa Ontario, Canada
No rounding to .176 gives 48.69428.....
So 48.4 looks fine to me.
 

Denis

Math Team
Oct 2011
14,592
1,026
Ottawa Ontario, Canada
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).
 

skeeter

Math Team
Jul 2011
3,355
1,848
Texas
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
$\tan(10^\circ) = \dfrac{h}{900} \implies h = 900\tan(10^\circ) = 900(0.176) = 158.4$

$158.4 - 110 = 48.4$

what typo?
 
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May 2008
302
81
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.
 
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