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Difference between revisions of "2024 AMC 12A Problems/Problem 20"

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<cmath>[\Delta APQ]=\dfrac12\cdot x\cdot y\sin(\angle PAQ)=\dfrac{xy}2\sin(60^\circ)=\dfrac{\sqrt3}4xy.</cmath>  
 
<cmath>[\Delta APQ]=\dfrac12\cdot x\cdot y\sin(\angle PAQ)=\dfrac{xy}2\sin(60^\circ)=\dfrac{\sqrt3}4xy.</cmath>  
  
−
Without loss of generality, we let <math>AB=BC=CA=1</math>, and thus <math>[\Delta ABC]=\dfrac{\sqrt3}4</math>; we therefore require <math>\dfrac{\sqrt3}4xy\le\dfrac12\cdot\dfrac{\sqrt3}4\implies xy\le\dfrac12</math> for <math>0\le x,y\le1</math>. A quick rough sketch of <math>y=\dfrac1{2x}</math> on the square given by <math>x,y\in[0,1]</math> reveals that the curve intersects the boundaries at <math>(0.5,1)</math> and <math>(1,0.5)</math>, and it is actually quite (very) obvious that the area bounded by the inequality <math>xy\le0.5</math> and the aforementioned unit square is more than <math>\dfrac34</math> but less than <math>\dfrac78</math> (cf. the diagram below). Thus, our answer is <math>\boxed{\textbf{(D) }\left(\dfrac34,\dfrac78\right]}</math>.  
+
Without loss of generality, we let <math>AB=BC=CA=1</math>, and thus <math>[\Delta ABC]=\dfrac{\sqrt3}4</math>; we therefore require <math>\dfrac{\sqrt3}4xy\le\dfrac12\cdot\dfrac{\sqrt3}4\implies xy\le\dfrac12</math> for <math>0\le x,y\le1</math>. (Note: You can skip most of this by using triangle area ratios ~A_MatheMagician.)
 +
 
 +
A quick rough sketch of <math>y=\dfrac1{2x}</math> on the square given by <math>x,y\in[0,1]</math> reveals that the curve intersects the boundaries at <math>(0.5,1)</math> and <math>(1,0.5)</math>, and it is actually quite (very) obvious that the area bounded by the inequality <math>xy\le0.5</math> and the aforementioned unit square is more than <math>\dfrac34</math> but less than <math>\dfrac78</math> (cf. the diagram below). Thus, our answer is <math>\boxed{\textbf{(D) }\left(\dfrac34,\dfrac78\right]}</math>.  
  
 
~Technodoggo
 
~Technodoggo
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draw((0.5,1)--(1,0.5),red+dashed+1.1);
 
draw((0.5,1)--(1,0.5),red+dashed+1.1);
 
</asy>
 
</asy>
 +
 
== Solution 2 ==
 
== Solution 2 ==
 
WLOG let <math>AB=AC=1</math>
 
WLOG let <math>AB=AC=1</math>
−
<cmath> \frac{AP \cdot AQ \cdot sin60}{2} < \frac{1 \cdot 1 \cdot sin60}{4}</cmath>
+
<cmath> \frac{AP \cdot AQ \cdot \sin60}{2} < \frac{1 \cdot 1 \cdot \sin60}{4}</cmath>
 
<cmath>AP \cdot AQ < \frac{1}{2}</cmath>
 
<cmath>AP \cdot AQ < \frac{1}{2}</cmath>
−
Which we can express as <math>xy < \frac{1}{2}</math> for graphing purposes <math>(x,y<1)</math>}
+
Which we can express as <math>xy < \frac{1}{2}</math>.
−
By graphing it out (someone please insert diagram)}
+
 
−
We see that the probability is slighty less than <math>\frac{7}{8}</math> but definitely greater than <math>\frac{3}{4}</math>
+
We follow by graphing the equation <math>xy < \frac 1 2</math> on <math>x, y \in [0, 1]</math>, and we follow the same way solution 1 does.
−
Thus answer choice <math>\boxed{(D) \left(\frac{3}{4},\frac{7}{8} \right]}</math>
+
 
−
Note: the actual probability can be found using integration
+
We see that the probability is slightly less than <math>\frac{7}{8}</math> but definitely greater than <math>\frac{3}{4}</math>
−
<cmath>P=\int_{0.5}^{1}{\frac{1}{2x}}dx+0.5= \frac{ln2+1}{2}\sim0.84655 < \frac{7}{8}</cmath>
+
 
 +
Thus answer choice <math>\boxed{\text{(D)} \left(\frac{3}{4},\frac{7}{8} \right]}</math>
 +
 
 +
==Solution 3==
 +
 
 +
The actual probability can be found by using calculus, and taking the integral.
 +
 
 +
<math>P=\int_{0.5}^{1}{\frac{1}{2x}}dx + 0.5= \frac{\ln2+1}{2}\approx 0.84657</math>.
 +
 
 +
Thus, since <math>\frac 3 4 < 0.84655 < \frac 7 8</math>, we get <math>\boxed{\text{(D)} \left(\frac{3}{4},\frac{7}{8} \right]}</math>
 
~lptoggled
 
~lptoggled
 +
 +
Note that if you do not know an approximation for <math>\ln2</math>, you can use the following power series formula: <cmath>\ln(1+x)=x-\frac{x^2}2+\frac{x^3}3-\frac{x^4}4+...</cmath> Using this formula to four terms gives the apprximation <math>\ln2 \approx \frac{19}{24}</math>, which is good enough to solve the problem.
 +
 +
===Another note===
 +
If you're feeling up to it, you can assume that the curve is actually a <math>90^{\circ}</math> sector of a circle with radius <math>\frac{1}{2}.</math> Doing so yields
 +
<cmath>P = 1 - \pi\frac{\left(\frac{1}{2}\right)^2}{4} \approx 0.8037,</cmath>
 +
which is clearly within the interval <math>D</math>. This is also an under-estimate, since the curve of <math>\frac{1}{2x}</math> is not as steep on the interval <math>x\in (0.5, 1).</math>
  
 
==See also==
 
==See also==
 
{{AMC12 box|year=2024|ab=A|num-b=19|num-a=21}}
 
{{AMC12 box|year=2024|ab=A|num-b=19|num-a=21}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 14:29, 17 November 2024

Problem

Points $P$ and $Q$ are chosen uniformly and independently at random on sides $\overline {AB}$ and $\overline{AC},$ respectively, of equilateral triangle $\triangle ABC.$ Which of the following intervals contains the probability that the area of $\triangle APQ$ is less than half the area of $\triangle ABC?$

$\textbf{(A) } \left[\frac 38, \frac 12\right] \qquad \textbf{(B) } \left(\frac 12, \frac 23\right] \qquad \textbf{(C) } \left(\frac 23, \frac 34\right] \qquad \textbf{(D) } \left(\frac 34, \frac 78\right] \qquad \textbf{(E) } \left(\frac 78, 1\right]$

Solution 1

Let $\overline{AP}=x$ and $\overline{AQ}=y$. Applying the sine formula for a triangle's area, we see that \[[\Delta APQ]=\dfrac12\cdot x\cdot y\sin(\angle PAQ)=\dfrac{xy}2\sin(60^\circ)=\dfrac{\sqrt3}4xy.\]

Without loss of generality, we let $AB=BC=CA=1$, and thus $[\Delta ABC]=\dfrac{\sqrt3}4$; we therefore require $\dfrac{\sqrt3}4xy\le\dfrac12\cdot\dfrac{\sqrt3}4\implies xy\le\dfrac12$ for $0\le x,y\le1$. (Note: You can skip most of this by using triangle area ratios ~A_MatheMagician.)

A quick rough sketch of $y=\dfrac1{2x}$ on the square given by $x,y\in[0,1]$ reveals that the curve intersects the boundaries at $(0.5,1)$ and $(1,0.5)$, and it is actually quite (very) obvious that the area bounded by the inequality $xy\le0.5$ and the aforementioned unit square is more than $\dfrac34$ but less than $\dfrac78$ (cf. the diagram below). Thus, our answer is $\boxed{\textbf{(D) }\left(\dfrac34,\dfrac78\right]}$.

~Technodoggo

[asy] /*Asymptote visual by Technodoggo, 7 November 2024*/ unitsize(8cm); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); label("$0$",(-0.05,-0.05)); label("$1$",(1,-0.05),S); label("$1$",(-0.05,1),W); draw((-0.05,0)--(1,0)--(1,-0.05)); draw((0,-0.05)--(0,1)--(-0.05,1)); real f(real x) {return 1/(2*x);} path c = graph(f, 0.5,1)--(1,0)--(0,0)--(0,1)--cycle; filldraw(c,blue+white); draw((0.5,1)--(0.5,0.5)--(1,0.5),white+dashed+1.1); draw((0.5,1)--(1,0.5),red+dashed+1.1); [/asy]

Solution 2

WLOG let $AB=AC=1$ \[\frac{AP \cdot AQ \cdot \sin60}{2} < \frac{1 \cdot 1 \cdot \sin60}{4}\] \[AP \cdot AQ < \frac{1}{2}\] Which we can express as $xy < \frac{1}{2}$.

We follow by graphing the equation $xy < \frac 1 2$ on $x, y \in [0, 1]$, and we follow the same way solution 1 does.

We see that the probability is slightly less than $\frac{7}{8}$ but definitely greater than $\frac{3}{4}$

Thus answer choice $\boxed{\text{(D)} \left(\frac{3}{4},\frac{7}{8} \right]}$

Solution 3

The actual probability can be found by using calculus, and taking the integral.

$P=\int_{0.5}^{1}{\frac{1}{2x}}dx + 0.5= \frac{\ln2+1}{2}\approx 0.84657$.

Thus, since $\frac 3 4 < 0.84655 < \frac 7 8$, we get $\boxed{\text{(D)} \left(\frac{3}{4},\frac{7}{8} \right]}$ ~lptoggled

Note that if you do not know an approximation for $\ln2$, you can use the following power series formula: \[\ln(1+x)=x-\frac{x^2}2+\frac{x^3}3-\frac{x^4}4+...\] Using this formula to four terms gives the apprximation $\ln2 \approx \frac{19}{24}$, which is good enough to solve the problem.

Another note

If you're feeling up to it, you can assume that the curve is actually a $90^{\circ}$ sector of a circle with radius $\frac{1}{2}.$ Doing so yields \[P = 1 - \pi\frac{\left(\frac{1}{2}\right)^2}{4} \approx 0.8037,\] which is clearly within the interval $D$. This is also an under-estimate, since the curve of $\frac{1}{2x}$ is not as steep on the interval $x\in (0.5, 1).$

See also

2024 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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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