Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2016 AMC 12B Problems/Problem 13"

(→‎Solution)
m (→‎Solution 2)
Line 26: Line 26:
  
  
−
Set the disance from Alice's and Bob's position to the point directly below the airplane to be <math>x</math> and <math>y</math>, respectively. From the Pythagorean Theorem, <math>x^2 + y^2 = 100</math>. As both are <math>30-60-90</math> triangles, the altitude of the airplane can be expressed as <math>\dfrac{x\sqrt{3}}{3}</math> or <math>y\sqrt{3}</math>. Solving the equation <math>\dfrac{x\sqrt{3}}{3} = y\sqrt{3}</math>, we get <math>x = 3y</math>. Plugging this into the equation <math>x^2 + y^2 = 100</math>, we get <math>10y^2 = 100</math>, or <math>y = \sqrt{10}</math> (<math>y</math> cannot be negative), so the altitude is <math>\sqrt{3*10} = \sqrt{30}</math>, which is closest to $\boxed{\textbf{B)}\ 5.5}.
+
Set the disance from Alice's and Bob's position to the point directly below the airplane to be <math>x</math> and <math>y</math>, respectively. From the Pythagorean Theorem, <math>x^2 + y^2 = 100</math>. As both are <math>30-60-90</math> triangles, the altitude of the airplane can be expressed as <math>\dfrac{x\sqrt{3}}{3}</math> or <math>y\sqrt{3}</math>. Solving the equation <math>\dfrac{x\sqrt{3}}{3} = y\sqrt{3}</math>, we get <math>x = 3y</math>. Plugging this into the equation <math>x^2 + y^2 = 100</math>, we get <math>10y^2 = 100</math>, or <math>y = \sqrt{10}</math> (<math>y</math> cannot be negative), so the altitude is <math>\sqrt{3*10} = \sqrt{30}</math>, which is closest to <math>\boxed{\textbf{B)}\ 5.5}.</math>
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2016|ab=B|num-b=12|num-a=14}}
 
{{AMC12 box|year=2016|ab=B|num-b=12|num-a=14}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 18:23, 28 February 2016

Problem

Alice and Bob live $10$ 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 $30^\circ$ from Alice's position and $60^\circ$ from Bob's position. Which of the following is closest to the airplane's altitude, in miles?

$\textbf{(A)}\ 3.5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 4.5 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 5.5$

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, $\tan(\theta)=\frac{z}{y}$. $\tan{30}=\frac{\sin{30}}{\cos{30}}=\frac{1}{\sqrt{3}}$. So, $y=z*\sqrt{3}$

From Bob's point of view, $\tan(\theta)=\frac{z}{x}$. $\tan{60}=\frac{\sin{60}}{\cos{60}}=\sqrt{3}$. So, $x = \frac{z}{\sqrt{3}}$

We know that $x^2$ + $y^2$ = $10^2$

Solving the equation (by plugging in x and y), we get z=$\sqrt{30}$ = about 5.5.

So, answer is $E) 5.5$

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 $x$ and $y$, respectively. From the Pythagorean Theorem, $x^2 + y^2 = 100$. As both are $30-60-90$ triangles, the altitude of the airplane can be expressed as $\dfrac{x\sqrt{3}}{3}$ or $y\sqrt{3}$. Solving the equation $\dfrac{x\sqrt{3}}{3} = y\sqrt{3}$, we get $x = 3y$. Plugging this into the equation $x^2 + y^2 = 100$, we get $10y^2 = 100$, or $y = \sqrt{10}$ ($y$ cannot be negative), so the altitude is $\sqrt{3*10} = \sqrt{30}$, which is closest to $\boxed{\textbf{B)}\ 5.5}.$

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2016_AMC_12B_Problems/Problem_13&oldid=77103"