Trigonometry -- Airport Word Problems

Date: 6/6/96 at 20:55:49
From: Anonymous
Subject: Trigonometry - word problems

1) A plane is 120 miles north and 85 miles east of an airport. If the 
pilot wants to fly directly to the airport, what bearing should be 
taken? 

2) When an airplane leaves the runway, its angle of climb is 18 
degrees and its speed is 275 feet per second. Find the altitude of the 
plane after 1 minute.

Date: 6/7/96 at 7:54:59
From: Doctor Anthony
Subject: Re: Trigonometry - word problems

(1) Draw a right-angled triangle ABC with A the airport.  B is due 
north of A and AB = 120 miles.  C is due east of B and BC = 85 miles.  
We wish to calculate angle BAC.  With this as our reference angle, AB 
is the 'adjacent' side and BC is the 'opposite' side.  Then we use the 
trig. ratio 'tan' to write down the equation

  tan(BAC) = opposite/adjacent

           = BC/AB = 85/120 = 0.708333

To find the angle whose 'tan' is 0.708333 we use the INVERSE tan 
function, and this gives angle(BAC) = 35.31 degrees

This is the direction going from the airport to the aircraft.  
However, we want the direction from the aircraft to the airport. This 
is the reciprocal of 35.31, and is found by adding 180 degrees, so the 
direction from aircraft to airport is 35.31 + 180 = 215.31 degrees.

(2) Draw another right-angled triangle ABC.  A is the point where the 
aircraft leaves the runway and AB is the path the aircraft follows, 
sloping at 18 degrees to the horizontal.  Since it travels at 275 ft/
sec and climbs for 60 seconds the distance AB is 275*60 = 16500 ft.  
BC is the perpendicular from B to the ground, and is the height we 
wish to calculate. Taking angle BAC (=18 degrees) as the reference 
angle, then BC is the 'opposite' side and AB the hypotenuse of the 
right-angled triangle ABC.

We can use the 'sin' function to write down an equation connecting the 
sides BC, AB and the angle BAC.

BC/AB = sin(18) = 0.309017

    BC = AB*0.309017

    BC = 16500*0.309017

    BC = 5098.78 feet

So the aircraft is at a height of 5099 feet after one minute.

-Doctor Anthony,  The Math Forum
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