2016 AMC 12B Problems/Problem 13
Contents
Problem
Alice and Bob live 
miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is 
from Alice's position and 
from Bob's position. Which of the following is closest to the airplane's altitude, in miles?
Solution
Let's set the altitude = z, distance from Alice to airplane's ground position (point right below airplane)=y and distance from Bob to airplane's ground position=x
From Alice's point of view, 
. 
. So, 
From Bob's point of view, 
. 
. So, 
We know that 
+ 
= 
Solving the equation (by plugging in x and y), we get z=
= about 5.5.
So, answer is 
solution by sudeepnarala
Solution 2
Non-trig solution by e_power_pi_times_i
Set the disance from Alice's and Bob's position to the point directly below the airplane to be 
and 
, respectively. From the Pythagorean Theorem, 
. As both are 
triangles, the altitude of the airplane can be expressed as 
or 
. Solving the equation 
, we get 
. Plugging this into the equation , we get 
, or 
( cannot be negative), so the altitude is 
, which is closest to 
See Also
| 2016 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 12 |
Followed by Problem 14 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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