Difference between revisions of "1993 AIME Problems/Problem 13"

Latest revision as of 03:38, 27 August 2024

Problem

Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let $t\,$ be the amount of time, in seconds, before Jenny and Kenny can see each other again. If $t\,$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

Solution

Solution 1

Consider the unit cicle of radius 50. Assume that they start at points $(-50,100)$ and $(-50,-100).$ Then at time $t$ , they end up at points $(-50+t,100)$ and $(-50+3t,-100).$ The equation of the line connecting these points and the equation of the circle are $\begin{align}y&=-\frac{100}{t}x+200-\frac{5000}{t}\\50^2&=x^2+y^2\end{align}.$ When they see each other again, the line connecting the two points will be tangent to the circle at the point $(x,y).$ Since the radius is perpendicular to the tangent we get $-\frac{x}{y}=-\frac{100}{t}$ or $xt=100y.$ Now substitute $y= \frac{xt}{100}$ into $(2)$ and get $x=\frac{5000}{\sqrt{100^2+t^2}}.$ Now substitute this and $y=\frac{xt}{100}$ into $(1)$ and solve for $t$ to get $t=\frac{160}{3}.$ Finally, the sum of the numerator and denominator is $160+3=\boxed{163}.$

Solution 2

Let $A$ and $B$ be Kenny's initial and final points respectively and define $C$ and $D$ similarly for Jenny. Let $O$ be the center of the building. Also, let $X$ be the intersection of $AC$ and $BD$ . Finaly, let $P$ and $Q$ be the points of tangency of circle $O$ to $AC$ and $BD$ respectively.

$[asy] size(8cm); defaultpen(linewidth(0.7)); pair A,B,C,D,P,Q,O,X; A=(0,0); B=(0,160); C=(200,0); D=(200,53.333); P=(100,0); Q=(123.529,94.118); O=(100,50); X=(300,0); dot(A); dot(B); dot(C); dot(D); dot(P); dot(Q); dot(O); dot(X); draw(A--B--X--cycle); draw(C--D); draw(P--O--Q); draw(O--X); draw(Circle(O,50)); label("$A$",A,SW); label("$B$",B,NNW); label("$C$",C,S); label("$D$",D,NE); label("$P$",P,S); label("$Q$",Q,NE); label("$O$",O,W); label("$X$",X,ESE); [/asy]$

From the problem statement, $AB=3t$ , and $CD=t$ . Since $\Delta ABX \sim \Delta CDX$ , $CX=AC\cdot\left(\frac{CD}{AB-CD}\right)=200\cdot\left(\frac{t}{3t-t}\right)=100$ .

Since $PC=100$ , $PX=200$ . So, $\tan(\angle OXP)=\frac{OP}{PX}=\frac{50}{200}=\frac{1}{4}$ .

Since circle $O$ is tangent to $BX$ and $AX$ , $OX$ is the angle bisector of $\angle BXA$ .

Thus, $\tan(\angle BXA)=\tan(2\angle OXP)=\frac{2\tan(\angle OXP)}{1- \tan^2(\angle OXP)} = \frac{2\cdot \left(\frac{1}{4}\right)}{1-\left(\frac{1}{4}\right)^2}=\frac{8}{15}$ .

Therefore, $t = CD = CX\cdot\tan(\angle BXA) = 100 \cdot \frac{8}{15} = \frac{160}{3}$ , and the answer is $\boxed{163}$ .

Solution 3

$[asy] size(8cm); defaultpen(linewidth(0.7)); pair A,B,C,D,P,Q,O,R,S; A=(0,0); B=(0,160); C=(200,0); D=(200,53.333); P=(100,0); Q=(123.529,94.118); O=(100,50); R=(100,106.667); S=(0,53.333); dot(A); dot(B); dot(C); dot(D); dot(P); dot(Q); dot(O); dot(R); dot(S); draw(A--B--D--C--cycle); draw(P--O); draw(D--S); draw(O--Q--R--cycle); draw(Circle(O,50)); label("$A$",A,SW); label("$B$",B,NNW); label("$C$",(200,-205),S); label("$D$",D,NE); label("$P$",(100,-205),S); label("$Q$",Q,NE); label("$O$",O,SW); label("$R$",R,NE); label("$S$",S,W); [/asy]$

Let $t$ be the time they walk. Then $CD=t$ and $AB=3t$ .

Draw a line from point $O$ to $Q$ such that $OQ$ is perpendicular to $BD$ . Further, draw a line passing through points $O$ and $P$ , so $OP$ is parallel to $AB$ and $CD$ and is midway between those two lines. Then $PR=\dfrac{AB+CD}{2}=\dfrac{3t+t}{2}=2t$ . Draw another line passing through point $D$ and parallel to $AC$ , and call the point of intersection of this line with $AB$ as $S$ . Then $SB=AB-AS=3t-t=2t$ .

We see that $m\angle SBD=m\angle ORQ$ since they are corresponding angles, and thus by angle-angle similarity, $\triangle QOR\sim\triangle SDB$ .

Then $\begin{align*} \dfrac{OQ}{DS}=\dfrac{RO}{BD}&\implies\dfrac{50}{200}=\dfrac{RO}{\sqrt{200^2+4t^2}}\\ &\implies RO=\dfrac{1}{4}\left(\sqrt{200^2+4t^2}\right)\\ &\implies RO=\dfrac{1}{2}\left(\sqrt{100^2+t^2}\right) \end{align*}$

And we obtain $\begin{align*} PR-OP&=RO\\ 2t-50&=\dfrac{1}{2}\left(\sqrt{100^2+t^2}\right)\\ 4t-100&=\sqrt{100^2+t^2}\\ (4t-100)^2&=\left(\sqrt{100^2+t^2}\right)^2\\ 16t^2-800t+100^2&=t^2+100^2\\ 15t^2&=800t\\ t&=\dfrac{800}{15} \end{align*}$

so we have $t=\frac{160}{3}$ , and our answer is thus $160+3=\boxed{163}$ .

Solution 4

We can use areas to find the answer. Since Jenny and Kenny are 200 feet apart, we know that they are side by side, and that the line connecting the two of them is tangent to the circular building.

We know that the radius of the circle is 50, and that $\overline{AJ} = x$ , $\overline{BK} = 3x$ .

Illustration:

$[asy] size(8cm); pair A = (0,0), B = (200,0), K = (200,160), J = (0,53.3333), O = (100,50); pair D = tangent(J,O,50,2), F = tangent(A,O,50); pair[] p = {A,B,K,J,O,D,F}; for (pair point : p) { dot(point); } label("A",A,SW); label("B",B,SE); label("K",K,NE); label("J",J,NW); label("O",O,SW); label("D",D,N); label("F",F,S); draw(Circle(O,50)); draw(A--B--K--J--cycle); draw(F--O--J); draw(D--O); draw(O--K); label("$50$",(100,26.666),0.5E); label("$50$",midpoint(D--O),0.5NE); label("$x$",midpoint(A--J),0.5W); label("$3x$",midpoint(B--K),0.5E); label("$200$",midpoint(A--B),4S); draw(rightanglemark(J,A,F,150)); draw(rightanglemark(J,D,O,150)); draw(rightanglemark(A,F,O,150)); draw(rightanglemark(F,B,K,150)); [/asy]$

By areas, $[OJK] + [AJOF] + [OFBK] = [ABKJ]$ . Having right trapezoids, $[AFOJ] = \frac{x+50}{2} \cdot 100$ . The other areas of right trapezoids can be calculated in the same way. We just need to find $[OJK]$ in terms of $x$ .

If we bring $\overline{AB}$ up to where the point J is, we have by the Pythagorean Theorem, $\overline{JK} = 2\sqrt{x^2+10000} \Rightarrow [OJK] = \frac{1}{2} \overline{JK} \cdot \overline{OD}$ .

Now we have everything to solve for $x$ .

$[OJK] + [AJOF] + [OFBK] = [ABKJ]$

$50 \sqrt{x^2 + 10000} + \frac{x+50}{2} \cdot 100 + \frac{3x+50}{2} \cdot 100 = \frac{x+3x}{2} \cdot 200$

After isolating the radical, dividing by 50, and squaring, we obtain: $15x^2 - 800x = 0 \Rightarrow x = \frac{160}{3}$ .

Since Jenny walks $\frac{160}{3}$ feet at 1 foot/s, our answer is $160+3 = \fbox{163}$ .

Solution 5

Let the point where Jenny started be $A = (0,0)$ . Then let $D = (t,0)$ the point Jenny is when she can see Kenny. Similarly, let $B = (0,200)$ be where Kenny started, and let $C = (3t,200)$ the point Kenny is when she can see Jenny. Then the equation of the line passing through $C, D$ is $100x-ty-100t=0$ . Now, note that the center of the building is at $O = (50,100)$ , and the distance from $O$ to $CD$ is $50$ , so, using the distance from a point to a line formula, we get $\frac{|100\cdot50-100a-100a|}{\sqrt{10000+a^2}} = 50,$ which simplifies to $15a^2-800a = 0$ from clearing denominators, squaring to remove absolute value and radical, then cancelling and rearranging. But this is equivalent to $a = \frac{160}{3}$ , so our answer is $\boxed{163}$ .

~Yiyj1

@@ Line 2: / Line 2: @@
 Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let <math>t\,</math> be the amount of time, in seconds, before Jenny and Kenny can see each other again. If <math>t\,</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
-== Solution 1==
+__TOC__
+== Solution ==
+=== Solution 1===
 Consider the unit cicle of radius 50. Assume that they start at points <math>(-50,100)</math> and <math>(-50,-100).</math> Then at time <math>t</math>, they end up at points <math>(-50+t,100)</math> and <math>(-50+3t,-100).</math>
 The equation of the line connecting these points and the equation of the circle are <cmath>\begin{align}y&=-\frac{100}{t}x+200-\frac{5000}{t}\\50^2&=x^2+y^2\end{align}.</cmath> When they see each other again, the line connecting the two points will be tangent to the circle at the point <math>(x,y).</math> Since the radius is perpendicular to the tangent we get <cmath>-\frac{x}{y}=-\frac{100}{t}</cmath> or <math>xt=100y.</math> Now substitute <cmath>y= \frac{xt}{100}</cmath> into <math>(2)</math> and get <cmath>x=\frac{5000}{\sqrt{100^2+t^2}}.</cmath> Now substitute this and <cmath>y=\frac{xt}{100}</cmath> into <math>(1)</math> and solve for <math>t</math> to get <cmath>t=\frac{160}{3}.</cmath> Finally, the sum of the numerator and denominator is <math>160+3=\boxed{163}.</math>
-==Solution 2==
+===Solution 2===
 Let <math>A</math> and <math>B</math> be Kenny's initial and final points respectively and define <math>C</math> and <math>D</math> similarly for Jenny. Let <math>O</math> be the center of the building. Also, let <math>X</math> be the intersection of <math>AC</math> and <math>BD</math>. Finaly, let <math>P</math> and <math>Q</math> be the points of tangency of circle <math>O</math> to <math>AC</math> and <math>BD</math> respectively.
-<asy>
+<center><asy>
-size(8cm);
+size(8cm); defaultpen(linewidth(0.7));
 pair A,B,C,D,P,Q,O,X;
 A=(0,0); B=(0,160); C=(200,0); D=(200,53.333); P=(100,0); Q=(123.529,94.118); O=(100,50); X=(300,0);
@@ Line 21: / Line 23: @@
 draw(Circle(O,50));
 label("$A$",A,SW); label("$B$",B,NNW); label("$C$",C,S); label("$D$",D,NE); label("$P$",P,S); label("$Q$",Q,NE); label("$O$",O,W); label("$X$",X,ESE);
-</asy>
+</asy></center>
 From the problem statement, <math>AB=3t</math>, and <math>CD=t</math>. Since <math>\Delta ABX \sim \Delta CDX</math>, <math>CX=AC\cdot\left(\frac{CD}{AB-CD}\right)=200\cdot\left(\frac{t}{3t-t}\right)=100</math>.
@@ Line 31: / Line 33: @@
 Thus, <math>\tan(\angle BXA)=\tan(2\angle OXP)=\frac{2\tan(\angle OXP)}{1- \tan^2(\angle OXP)} = \frac{2\cdot \left(\frac{1}{4}\right)}{1-\left(\frac{1}{4}\right)^2}=\frac{8}{15}</math>.
 Therefore, <math>t = CD = CX\cdot\tan(\angle BXA) = 100 \cdot \frac{8}{15} = \frac{160}{3}</math>, and the answer is <math>\boxed{163}</math>.
+===Solution 3===
+<asy>
+size(8cm); defaultpen(linewidth(0.7));
+pair A,B,C,D,P,Q,O,R,S;
+A=(0,0); B=(0,160); C=(200,0); D=(200,53.333); P=(100,0); Q=(123.529,94.118); O=(100,50); R=(100,106.667); S=(0,53.333);
+dot(A); dot(B); dot(C); dot(D); dot(P); dot(Q); dot(O); dot(R); dot(S);
+draw(A--B--D--C--cycle);
+draw(P--O); draw(D--S);
+draw(O--Q--R--cycle);
+draw(Circle(O,50));
+label("$A$",A,SW); label("$B$",B,NNW); label("$C$",(200,-205),S); label("$D$",D,NE); label("$P$",(100,-205),S); label("$Q$",Q,NE); label("$O$",O,SW); label("$R$",R,NE); label("$S$",S,W);
+</asy>
+Let <math>t</math> be the time they walk.  Then <math>CD=t</math> and <math>AB=3t</math>.
+Draw a line from point <math>O</math> to <math>Q</math> such that <math>OQ</math> is perpendicular to <math>BD</math>.  Further, draw a line passing through points <math>O</math> and <math>P</math>, so <math>OP</math> is parallel to <math>AB</math> and <math>CD</math> and is midway between those two lines.  Then <math>PR=\dfrac{AB+CD}{2}=\dfrac{3t+t}{2}=2t</math>.  Draw another line passing through point <math>D</math> and parallel to <math>AC</math>, and call the point of intersection of this line with <math>AB</math> as <math>S</math>.  Then <math>SB=AB-AS=3t-t=2t</math>.
+We see that <math>m\angle SBD=m\angle ORQ</math> since they are corresponding angles, and thus by angle-angle similarity, <math>\triangle QOR\sim\triangle SDB</math>.
+Then
+<cmath>\begin{align*}
+\dfrac{OQ}{DS}=\dfrac{RO}{BD}&\implies\dfrac{50}{200}=\dfrac{RO}{\sqrt{200^2+4t^2}}\\
+&\implies RO=\dfrac{1}{4}\left(\sqrt{200^2+4t^2}\right)\\
+&\implies RO=\dfrac{1}{2}\left(\sqrt{100^2+t^2}\right)
+\end{align*}</cmath>
+And we obtain
+<cmath>\begin{align*}
+PR-OP&=RO\\
+t-50&=\dfrac{1}{2}\left(\sqrt{100^2+t^2}\right)\\
+t-100&=\sqrt{100^2+t^2}\\
+(4t-100)^2&=\left(\sqrt{100^2+t^2}\right)^2\\
+t^2-800t+100^2&=t^2+100^2\\
+t^2&=800t\\
+t&=\dfrac{800}{15}
+\end{align*}</cmath>
+so we have <math>t=\frac{160}{3}</math>, and our answer is thus <math>160+3=\boxed{163}</math>.
+===Solution 4===
+We can use areas to find the answer. Since Jenny and Kenny are 200 feet apart, we know that they are side by side, and that the line connecting the two of them is tangent to the circular building.
+We know that the radius of the circle is 50, and that <math>\overline{AJ} = x</math>, <math>\overline{BK} = 3x</math>.
+Illustration:
+<asy>
+    size(8cm);
+    pair A = (0,0), B = (200,0), K = (200,160), J = (0,53.3333), O = (100,50);
+    pair D = tangent(J,O,50,2), F = tangent(A,O,50);
+    pair[] p = {A,B,K,J,O,D,F};
+    for (pair point : p) {
+        dot(point);
+    }
+    label("A",A,SW);
+    label("B",B,SE);
+    label("K",K,NE);
+    label("J",J,NW);
+    label("O",O,SW);
+    label("D",D,N);
+    label("F",F,S);
+    draw(Circle(O,50));
+    draw(A--B--K--J--cycle);
+    draw(F--O--J);
+    draw(D--O);
+    draw(O--K);
+    label("$50$",(100,26.666),0.5E);
+    label("$50$",midpoint(D--O),0.5NE);
+    label("$x$",midpoint(A--J),0.5W);
+    label("$3x$",midpoint(B--K),0.5E);
+    label("$200$",midpoint(A--B),4S);
+    draw(rightanglemark(J,A,F,150));
+    draw(rightanglemark(J,D,O,150));
+    draw(rightanglemark(A,F,O,150));
+    draw(rightanglemark(F,B,K,150));
+</asy>
+By areas, <math>[OJK] + [AJOF] + [OFBK] = [ABKJ]</math>. Having right trapezoids, <math>[AFOJ] = \frac{x+50}{2} \cdot 100</math>. The other areas of right trapezoids can be calculated in the same way. We just need to find <math>[OJK]</math> in terms of <math>x</math>.
+If we bring <math>\overline{AB}</math> up to where the point J is, we have by the Pythagorean Theorem, <math>\overline{JK} = 2\sqrt{x^2+10000} \Rightarrow [OJK] = \frac{1}{2} \overline{JK} \cdot \overline{OD}</math>.
+Now we have everything to solve for <math>x</math>.
+<cmath>[OJK] + [AJOF] + [OFBK] = [ABKJ]</cmath>
+<cmath>50 \sqrt{x^2 + 10000} + \frac{x+50}{2} \cdot 100 + \frac{3x+50}{2} \cdot 100 = \frac{x+3x}{2} \cdot 200</cmath>
+After isolating the radical, dividing by 50, and squaring, we obtain: <math>15x^2 - 800x = 0 \Rightarrow x = \frac{160}{3}</math>.
+Since Jenny walks <math>\frac{160}{3}</math> feet at 1 foot/s, our answer is <math>160+3 = \fbox{163}</math>.
+==Solution 5==
+Let the point where Jenny started be <math>A = (0,0)</math>. Then let <math>D = (t,0)</math> the point Jenny is when she can see Kenny. Similarly, let <math>B = (0,200)</math> be where Kenny started, and let <math>C = (3t,200)</math> the point Kenny is when she can see Jenny. Then the equation of the line passing through <math>C, D</math> is <math>100x-ty-100t=0</math>. Now, note that the center of the building is at <math>O = (50,100)</math>, and the distance from <math>O</math> to <math>CD</math> is <math>50</math>, so, using the distance from a point to a line formula, we get <cmath>\frac{|100\cdot50-100a-100a|}{\sqrt{10000+a^2}} = 50,</cmath>
+which simplifies to <math>15a^2-800a = 0</math> from clearing denominators, squaring to remove absolute value and radical, then cancelling and rearranging. But this is equivalent to <math>a = \frac{160}{3}</math>, so our answer is <math>\boxed{163}</math>.
+~[[User:Yiyj1|Yiyj1]]
 == See also ==
 {{AIME box|year=1993|num-b=12|num-a=14}}
+[[Category:Intermediate Geometry Problems]]
+{{MAA Notice}}

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