Altitude Problem: Approximate the altitude of the plane.

[hp_16]

I don't get this trig problem right here, I wanted to know if someone here could help me out with it...here it is.

"A plane is observed approaching your home and you assume its speed is 550 miles per hour. The angle of elevation to the plane is 16 degrees at one time and 57 degrees one minute later. Approximate the altitude of the plane."

Does it make sense? Can someone help me with this please..

 

[tkhunny]

Sure it makes sense, but only if you assume a constant altitude. You must have a really tall house.

Draw a right triangle. A nice wide and short one will suit best. The right angle should be on the ground. Put your house on the ground acute angle and the plane on the acute angle in the sky. The plane should be directly above the right angle. Label the house angle 16º.

Draw a horizontal line from the plane somewhat toward the house. Don't go too far, maybe half way. This is the new plane position. Drop a perpendicular from the new plane position to the base of the original right triangle. Label the new right angle. Finally, draw a line segment from the house to the new position of the plane. Lable this wider house angle 57º.

That's about it. If you know something about tangents, you're in business.

 

[galactus]

I've been trying to post a diagram, but it won't display. Go figure.

Anyway, there are variuos ways to tackle this problem.

First, how far does the plane travel in one minute given it flies at 550 mph?. That's a distance you can use in your calculations.

 

[soroban]

Re: Altitude Problem

Hello, hp_16!

tkhunny is absolutely correct . . .

A plane is observed approaching your home and you assume its speed is 550 mph.
The angle of elevation to the plane is 16° at one time and 57° one minute later.
Approximate the altitude of the plane.

      Figure 1

      P
      *
      |     *
      |           *
    y |                 *
      |                       *
      |                       :16°  *
      * - - - - - - - - - - - + - - - - - *
      A         55/6          B     x     H

The plane is at \(\displaystyle P:\;PA\,=\,y\) (miles)
\(\displaystyle \angle PHA\,=\,16^o\)


In one minute, the plane has flown: \(\displaystyle \frac{1}{60}\,\times\,550 \:=\:\frac{55}{6}\) miles to point Q.
. . \(\displaystyle PQ\,=\,AB\,=\,\frac{55}{6}\)
Draw \(\displaystyle QB\,\perp\,AH,\;BH \,=\,x\)

      Figure 2

      P         55/6          Q
      * - - - - - - - - - - - *
      |                       : *
      |                       :   *
    y |                     y :     *
      |                       :       *
      |                       :     57° *
      * - - - - - - - - - - - + - - - - - *
      A         55/6          B     x     H

In Figure 1: \(\displaystyle \:\tan16^o \,=\,\frac{h}{x\,+\,\frac{55}{6}}\;\;\Rightarrow\;\;h \:=\:\frac{(6x\,+\,55)\tan16^o}{6}\;\) [1]

In Figure 2: \(\displaystyle \:\tan57^o \,=\,\frac{h}{x}\;\;\Rightarrow\;\;h \:=\:x\cdot\tan57^o\;\) [2]


Equate [1] and [2] and solve for \(\displaystyle x.\)

Substitute into [2] and solve for \(\displaystyle h.\)

 

[TchrWill]

Alternatively:


..............55/5
......B................C
*................*
| *
| * *
y | *
| * *
| *
* - - - - - - - - - - - + - - - - - *A
.....D

In case the FIgure does not show correctly:

A = the observation point
B = the initial position of the plane
C = the distance of the plane from point B after 1 minute or 55/6 miles.
D = a point directly below the plane's initial position, being the altitude, BD = h.
Angle BAD = 16º
Angle CBA = 16º
Angle CAD = 57º
Angle CAB = 41º
Angle BCA = 123º
BA/sin(123) = (55/6)/sin41º
BA = (55/6)sin(123)/sin(41)
BD = h = BAsin(16)

 

[jwpaine]

There are a number of ways to find the Heigth. You could use right angle trig methods to find the upper vertical section above the point where the 16 degree line interests the vertical line segment, and then use the angles provided along with the law of sine to find the lower vertical section below the point where the 16 degree line interests the vertical line segment. The height will then be the upper + lower sections.... that's how I did it.

John.