Difference between revisions of "2016 AIME I Problems/Problem 9"

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Problem 9

Triangle $ABC$ has $AB=40,AC=31,$ and $\sin{A}=\frac{1}{5}$ . This triangle is inscribed in rectangle $AQRS$ with $B$ on $\overline{QR}$ and $C$ on $\overline{RS}$ . Find the maximum possible area of $AQRS$ .

Solution 1

Note that if angle $BAC$ is obtuse, it would be impossible for the triangle to inscribed in a rectangle. This can easily be shown by drawing triangle ABC, where $A$ is obtuse. Therefore, angle A is acute. Let angle $CAS=n$ and angle $BAQ=m$ . Then, $\overline{AS}=31\cos(n)$ and $\overline{AQ}=40\cos(m)$ . Then the area of rectangle $AQRS$ is $1240\cos(m)\cos(n)$ . By product-to-sum, $\cos(m)\cos(n)=\frac{1}{2}(\cos(m+n)+\cos(m-n))$ . $\cos(m+n)=\sin(90-m-n)=\sin(BAC)=\frac{1}{5}$ . The maximum possible value of $\cos(m-n)$ is 1, which occurs when $m=n$ . Thus the maximum possible value of $\cos(m)\cos(n)$ is $\frac{1}{2}(\frac{1}{5}+1)=\frac{3}{5}$ so the maximum possible area of $AQRS$ is $1240\times{\frac{3}{5}}=\fbox{744}$ .

Solution 2

As above, we note that angle $A$ must be acute. Therefore, let $A$ be the origin, and suppose that $Q$ is on the positive $x$ axis and $S$ is on the positive $y$ axis. We approach this using complex numbers. Let $w=\text{cis} A$ , and let $z$ be a complex number with $|z|=1$ , $\text{Arg}(z)\ge 0^\circ$ and $\text{Arg}(zw)\le90^\circ$ . Then we represent $B$ by $40z$ and $C$ by $31zw$ . The coordinates of $Q$ and $S$ depend on the real part of $40z$ and the imaginary part of $31zw$ . Thus $[AQRS]=\Re(40z)\cdot \Im(31zw)=1240\left(\frac{z+\overline{z}}{2}\right)\left(\frac{zw-\overline{zw}}{2i}\right).$ We can expand this, using the fact that $z\overline{z}=|z|^2$ , finding $[AQRS]=620\left(\frac{z^2w-\overline{z^2w}+w-\overline{w}}{2i}\right)=620(\Im(z^2w)+\Im(w)).$ Now as $w=\text{cis}A$ , we know that $\Im(w)=\frac15$ . Also, $|z^2w|=1$ , so the maximum possible imaginary part of $z^2w$ is $1$ . This is clearly achievable under our conditions on $z$ . Therefore, the maximum possible area of $AQRS$ is $620(1+\tfrac15)=\boxed{744}$ .

Solution 3 (With Calculus)

Let $\theta$ be the angle $\angle BAQ$ . The height of the rectangle then can be expressed as $h = 31 \sin (A+\theta)$ , and the length of the rectangle can be expressed as $l = 40\cos \theta$ . The area of the rectangle can then be written as a function of $\theta$ , $[AQRS] = a(\theta) = 31\sin (A+\theta)\cdot 40 \cos \theta = 1240 \sin (A+\theta) \cos \theta$ . For now, we will ignore the $1240$ and focus on the function $f(\theta) = \sin (A+\theta) \cos \theta = (\sin A \cos \theta + \cos A \sin \theta)(\cos \theta) = \sin A \cos^2 \theta + \cos A \sin \theta \cos \theta = \sin A \cos^2 \theta + \frac{1}{2} \cos A \sin 2\theta$ .

Taking the derivative, $f'(\theta) = \sin A \cdot -2\cos \theta \sin \theta + \cos A \cos 2\theta = \cos A \cos 2\theta - \sin A \sin 2\theta = \cos(2\theta + A)$ . Setting this equal to $0$ , we get $\cos(2 \theta + A) = 0 \Rightarrow 2\theta +A = 90, 270 ^\circ$ . Since we know that $A+ \theta < 90$ , the $270^\circ$ solution is extraneous. Thus, we get that $\theta = \frac{90 - A}{2} = 45 - \frac{A}{2}$ .

Plugging this value into the original area equation, $a(45 - \frac{A}{2}) = 1240 \sin (45 - \frac{A}{2} + A) \cos (45 - \frac{A}{2}) = 1240\sin( 45+ \frac{A}{2})\cos(45 - \frac{A}{2})$ . Using a product-to-sum formula, we get that: $1240\sin( 45+ \frac{A}{2})\cos(45 - \frac{A}{2}) =$ $1240\cdot \frac{1}{2}\cdot(\sin((45 + \frac{A}{2}) + (45 -\frac{A}{2}))+\sin((45 +\frac{A}{2})-(45 - \frac{A}{2})))=$ $620 (\sin 90^\circ + \sin A) = 620 \cdot \frac{6}{5} = \boxed{744}$ .

Solution 4

Let $\alpha$ be the angle $\angle CAS$ and $\beta$ be the angle $\angle BAQ$ . Then $\alpha + \beta + \angle A = 90^\circ \Rightarrow \alpha + \beta = 90^\circ - \angle A$ $\cos(\alpha + \beta) = \cos(90^\circ - \angle A)$ $\cos(\alpha + \beta) = \sin(\angle A) = \frac{1}{5}$ $\cos\alpha\cos\beta - \sin\alpha\sin\beta = \frac{1}{5}$ $\cos\alpha\cos\beta - \sqrt{(1-\cos^2\alpha)(1-\cos^2\beta)} = \frac{1}{5}$ $\cos\alpha\cos\beta - \sqrt{1-\cos^2\alpha-\cos^2\beta+\cos^2\alpha\cos^2\beta} = \frac{1}{5}$ However, by AM-GM: $\cos^2\alpha+\cos^2\beta \ge 2\cos\alpha\beta$ Therefore, $1-\cos^2\alpha-\cos^2\beta+\cos^2\alpha\cos^2\beta \le 1-2\cos\alpha\beta+\cos^2\alpha\cos^2\beta = (1-\cos\alpha\cos\beta)^2$ $\sqrt{1-\cos^2\alpha-\cos^2\beta+\cos^2\alpha\cos^2\beta} \le 1-\cos\alpha\cos\beta$ So, $\frac{1}{5} \ge \cos\alpha\cos\beta - (1-\cos\alpha\cos\beta) = 2\cos\alpha\cos\beta-1$ $\frac{3}{5} \ge \cos\alpha\cos\beta$ . However, the area of the rectangle is just $AS \cdot AQ = 31\cos\alpha \cdot 40\cos\beta \le 31 \cdot 40 \cdot \frac{3}{5} = \boxed{744}$ .

Note on Problem Validity

It has been noted that this answer won't actually work. Let angle $QAB = m$ and angle $CAS = n$ as in Solution 1. Since we know (through that solution) that $m = n$ , we can call them each $\theta$ . The height of the rectangle is $AS = 31\cos\theta$ , and the distance $BQ = 40\sin\theta$ . We know that, if the triangle is to be inscribed in a rectangle, $AS \geq BQ$ .

$AS \geq BQ$

$31\cos\theta \geq 40\sin\theta$

$\frac{31}{40} \geq \tan\theta$

However, $\tan\theta = \tan(\frac{90-A}{2}) = \frac{\sin(90-A)}{\cos(90-A)+1} = \frac{\cos A}{\sin A + 1} = \frac{\frac{2\sqrt6}{5}}{\frac{6}{5}} = \frac{\sqrt6}{3} > \frac{31}{40}$ , so the triangle does not actually fit in the rectangle: specifically, B is above R and thus in the line containing segment QR but not on the actual segment or in the rectangle.

$[asy] size(200); pair A,B,C,Q,R,S; real r = (pi/2 - asin(1/5))/2; A = (0,0); B = 40*dir(r*180/pi); C = 31*dir(90-r*180/pi); draw(A--B--C--cycle); Q = (40*cos(r),0); R = (40*cos(r),31*cos(r)); S = (0, 31*cos(r)); draw(A--Q--R--S--cycle); label("$A$",A,SW); label("$B$",B,NE); label("$C$",C,N); label("$Q$",Q,SE); label("$R$",R,E); label("$S$",S,NW); [/asy]$

The actual answer is a radical near $728$ (letting the triangle be inside the rectangle). The CAMC, however, has decided to accept only the answer $744$ despite the invalid problem statement.

@@ Line 1: / Line 1: @@
 ==Problem 9==
 Triangle <math>ABC</math> has <math>AB=40,AC=31,</math> and <math>\sin{A}=\frac{1}{5}</math>. This triangle is inscribed in rectangle <math>AQRS</math> with <math>B</math> on <math>\overline{QR}</math> and <math>C</math> on <math>\overline{RS}</math>. Find the maximum possible area of <math>AQRS</math>.
-===Solution 1===
+==Solution 1==
 Note that if angle <math>BAC</math> is obtuse, it would be impossible for the triangle to inscribed in a rectangle. This can easily be shown by drawing triangle ABC, where <math>A</math> is obtuse. Therefore, angle A is acute. Let angle <math>CAS=n</math> and angle <math>BAQ=m</math>. Then, <math>\overline{AS}=31\cos(n)</math> and <math>\overline{AQ}=40\cos(m)</math>. Then the area of rectangle <math>AQRS</math> is <math>1240\cos(m)\cos(n)</math>. By product-to-sum, <math>\cos(m)\cos(n)=\frac{1}{2}(\cos(m+n)+\cos(m-n))</math>. <math>\cos(m+n)=\sin(90-m-n)=\sin(BAC)=\frac{1}{5}</math>. The maximum possible value of <math>\cos(m-n)</math> is 1, which occurs when <math>m=n</math>. Thus the maximum possible value of <math>\cos(m)\cos(n)</math> is <math>\frac{1}{2}(\frac{1}{5}+1)=\frac{3}{5}</math> so the maximum possible area of <math>AQRS</math> is <math>1240\times{\frac{3}{5}}=\fbox{744}</math>.
-===Solution 2===
+==Solution 2==
 As above, we note that angle <math>A</math> must be acute. Therefore, let <math>A</math> be the origin, and suppose that <math>Q</math> is on the positive <math>x</math> axis and <math>S</math> is on the positive <math>y</math> axis. We approach this using complex numbers. Let <math>w=\text{cis} A</math>, and let <math>z</math> be a complex number with <math>|z|=1</math>, <math>\text{Arg}(z)\ge 0^\circ</math>  and <math>\text{Arg}(zw)\le90^\circ</math>. Then we represent <math>B</math> by <math>40z</math> and <math>C</math> by <math>31zw</math>. The coordinates of <math>Q</math> and <math>S</math> depend on the real part of <math>40z</math> and the imaginary part of <math>31zw</math>. Thus
 <cmath>[AQRS]=\Re(40z)\cdot \Im(31zw)=1240\left(\frac{z+\overline{z}}{2}\right)\left(\frac{zw-\overline{zw}}{2i}\right).</cmath>
@@ Line 11: / Line 12: @@
 Now as <math>w=\text{cis}A</math>, we know that <math>\Im(w)=\frac15</math>. Also, <math>|z^2w|=1</math>, so the maximum possible imaginary part of <math>z^2w</math> is <math>1</math>. This is clearly achievable under our conditions on <math>z</math>. Therefore, the maximum possible area of <math>AQRS</math> is <math>620(1+\tfrac15)=\boxed{744}</math>.
-===Solution 3 (With Calculus)===
+==Solution 3 (With Calculus)==
 Let <math>\theta</math> be the angle <math>\angle BAQ</math>. The height of the rectangle then can be expressed as <math>h = 31 \sin (A+\theta)</math>, and the length of the rectangle can be expressed as <math>l = 40\cos \theta</math>. The area of the rectangle can then be written as a function of <math>\theta</math>, <math>[AQRS] = a(\theta) = 31\sin (A+\theta)\cdot 40 \cos \theta = 1240 \sin (A+\theta) \cos \theta</math>. For now, we will ignore the <math>1240</math> and focus on the function <math>f(\theta) = \sin (A+\theta) \cos \theta = (\sin A \cos \theta + \cos A \sin \theta)(\cos \theta) = \sin A \cos^2 \theta + \cos A \sin \theta \cos \theta = \sin A \cos^2 \theta + \frac{1}{2} \cos A \sin 2\theta</math>.
@@ Line 20: / Line 21: @@
 <cmath>620 (\sin 90^\circ + \sin A) = 620 \cdot \frac{6}{5} = \boxed{744}</cmath>.
-===Solution 4===
+==Solution 4==
 Let <math>\alpha</math> be the angle <math>\angle CAS</math> and <math>\beta</math> be the angle <math>\angle BAQ</math>. Then
 <cmath>\alpha + \beta + \angle A = 90^\circ \Rightarrow \alpha + \beta = 90^\circ - \angle A</cmath>
@@ Line 39: / Line 40: @@
-===Note on Problem Validity===
+==Note on Problem Validity==
 It has been noted that this answer won't actually work. Let angle <math>QAB = m</math> and angle <math>CAS = n</math> as in Solution 1. Since we know (through that solution) that <math>m = n</math>, we can call them each <math>\theta</math>. The height of the rectangle is <math>AS = 31\cos\theta</math>, and the distance <math>BQ = 40\sin\theta</math>. We know that, if the triangle is to be inscribed in a rectangle, <math>AS \geq BQ</math>.
@@ Line 71: / Line 72: @@
 label("$S$",S,NW);
 </asy>
 The actual answer is a radical near <math>728</math> (letting the triangle be inside the rectangle). The CAMC, however, has decided to accept only the answer <math>744</math> despite the invalid problem statement.
-=See Also=
+==See Also==
 {{AIME box|year=2016|n=I|num-b=8|num-a=10}}
 {{MAA Notice}}

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