Difference between revisions of "1989 AIME Problems/Problem 6"

Revision as of 17:00, 29 August 2019

Problem

Two skaters, Allie and Billie, are at points $A$ and $B$ , respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\circ$ angle with $AB$ . At the same time Allie leaves $A$ , Billie leaves $B$ at a speed of $7$ meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?

$[asy] pointpen=black; pathpen=black+linewidth(0.7); pair A=(0,0),B=(10,0),C=6*expi(pi/3); D(B--A); D(A--C,EndArrow); MP("A",A,SW);MP("B",B,SE);MP("60^{\circ}",A+(0.3,0),NE);MP("100",(A+B)/2); [/asy]$

Solution

Label the point of intersection as $C$ . Since $d = rt$ , $AC = 8t$ and $BC = 7t$ . According to the law of cosines,

$[asy] pointpen=black; pathpen=black+linewidth(0.7); pair A=(0,0),B=(10,0),C=16*expi(pi/3); D(B--A); D(A--C); D(B--C,dashed); MP("A",A,SW);MP("B",B,SE);MP("C",C,N);MP("60^{\circ}",A+(0.3,0),NE);MP("100",(A+B)/2);MP("8t",(A+C)/2,NW);MP("7t",(B+C)/2,NE); [/asy]$

$\begin{align*}(7t)^2 &= (8t)^2 + 100^2 - 2 \cdot 8t \cdot 100 \cdot \cos 60^\circ\\ 0 &= 15t^2 - 800t + 10000 = 3t^2 - 160t + 2000\\ t &= \frac{160 \pm \sqrt{160^2 - 4\cdot 3 \cdot 2000}}{6} = 20, \frac{100}{3}.\end{align*}$

Since we are looking for the earliest possible intersection, $20$ seconds are needed. Thus, $8 \cdot 20 = \boxed{160}$ meters is the solution.

Alternatively, we can drop an altitude from $C$ and arrive at the same answer.

Solution 2

Can someone please help me make an asymptote diagram for this?

$[asy] draw((0,0)--(11,0)--(7,14)--cycle); draw((7,14)--(11,28)); draw((11,28)--(11,0)); label("$A$",(-1,-1),N); label("$B$",(12,-1),N); label("$P$",(6,15),N); label("$B'$",(12,29),N); draw((-10,14)--(20,14)); label("$X$",(12,15),N); [/asy]$

dot((0,0)); label("A",(0,0),S); dot((100,0)); label("B",(100,0),SE); dot((80,139)); label("M",(80,139),N); draw((80,139)--(80,0),dashed); label("4x",(0,0)--(80,0),S); label("100-4x",(80,0)--(100,0),S); label(" $4x \sqrt{3}$ ",(80,139)--(80,0),W); label(" $60^{\circ}$ ", (3,1), NE);

Let the point of intersection be $P$ . We can draw a line that goes through $P$ and is parallel to $\overline{AB}$ . Letting this line be the $x$ axis, we can reflect $B$ over the $x$ axis to get $B'$ . Note that as reflections preserve length, $B'X = XB$ .

We then draw lines $BB'$ and $PB'$ . We can let the foot of the perpendicular from $P$ to $BB'$ be $X$ , and we can let the foot of the perpendicular from $P$ to $AB$ be $Y$ . In doing so, we have constructed rectangle $PXBY$ .

By $d=rt$ , we have $AP = 8t$ and $PB' = 7t$ . Furthermore, we have $30-60-90$ triangle $PAY$ , so $AY = 4t$ and $PY = 4t\sqrt{3}$ . Since we have $PY = XB = B'X$ , $B'X = 4t\sqrt{3}$ . By Pythagoras, $PX = t$ .

Then, we have $PX = BY$ . Since we know that $AY + YB = AB$ , we get $4t + t = 100$ . Solving for $t$ , we get $t = 20$ . Our answer is then equivalent to $8t$ . Thus, $8(20) = \boxed{160}$ meters is the solution. - Spacesam

Solution 3

We can define $x$ to be the time elapsed since both Allie and Billie moved away from points $A$ and $B$ respectfully. Also, set the point of intersection to be $M$ . Then we can produce the following diagram:

$[asy] draw((0,0)--(100,0)--(80,139)--cycle); label("8x",(0,0)--(80,139),NW); label("7x",(100,0)--(80,139),NE); label("100",(0,0)--(100,0),S); dot((0,0)); label("A",(0,0),S); dot((100,0)); label("B",(100,0),S); dot((80,139)); label("M",(80,139),N); [/asy]$

Now, if we drop an altitude from point $M$ , we get :

$[asy] size(300); draw((0,0)--(100,0)--(80,139)--cycle); label("8x",(0,0)--(80,139),NW); label("7x",(100,0)--(80,139),NE); dot((0,0)); label("A",(0,0),S); dot((100,0)); label("B",(100,0),SE); dot((80,139)); label("M",(80,139),N); draw((80,139)--(80,0),dashed); label("4x",(0,0)--(80,0),S); label("100-4x",(80,0)--(100,0),S); label("$4x \sqrt{3}$",(80,139)--(80,0),W); label("$60^{\circ}$", (3,1), NE); [/asy]$

We know this from the $30-60-90$ triangle that is formed. From this we get that: $(7x)^2 = (4 \sqrt{3})^2 + (100-4x)^2$

$\Longrightarrow 49x^2 - 48x^2 = x^2 = (100-4x)^2$

$\Longrightarrow 0=(100-4x)^2 - x^2 = (100-3x)(100-5x)$ .

Therefore, we get that $x = \frac{100}{3}$ or $x = 20$ . Since $20< \frac{100}{3}$ , we have that $x=20$ (since the problem asks for the quickest possible meeting point), so the distance Allie travels before meeting Billie would be $8 \cdot x = 8 \cdot 20 = \boxed{160}$ meters.

~qwertysri987

1989 AIME (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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