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Difference between revisions of "1962 AHSME Problems/Problem 37"

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<math> \textbf{(A)}\ \frac{1}2\qquad\textbf{(B)}\ \frac{9}{16}\qquad\textbf{(C)}\ \frac{19}{32}\qquad\textbf{(D)}\ \frac{5}{8}\qquad\textbf{(E)}\ \frac{2}3 </math>
 
<math> \textbf{(A)}\ \frac{1}2\qquad\textbf{(B)}\ \frac{9}{16}\qquad\textbf{(C)}\ \frac{19}{32}\qquad\textbf{(D)}\ \frac{5}{8}\qquad\textbf{(E)}\ \frac{2}3 </math>
  
 +
==Solution==
  
−
==Solution==
+
Let <math>AE=AF=x</math>
−
{{solution}}
+
<math>[CDFE]=[ABCD]-[AEF]-[EBC]=1-\frac{x^2}{2}-\frac{1-x}{2}</math>
 +
Or
 +
<math>[CDFE]=\frac{\frac{5}{4}-(x-\frac{1}{2})^2}{2}\le \frac{5}{8}</math>
 +
As <math>(x-\frac{1}{2})^2\ge 0</math>
 +
So <math>[CDFE]\le \frac{5}{8}</math>
 +
Equality occurs when <math>AE=AF=x=\frac{1}{2}</math>
 +
So the maximum value is <math>\frac{5}{8}</math>
 +
 
 +
==Solution 2==
 +
 
 +
Let us first draw a unit square.
 +
<asy>
 +
draw((0,0)--(0,1));
 +
draw((0,1)--(1,1));
 +
draw((1,1)--(1,0));
 +
draw((1,0)--(0,0));
 +
label("A",(0,1),NW);
 +
label("B",(1,1),NE);
 +
label("C",(1,0),SE);
 +
label("D",(0,0),SW);
 +
label("$1$",(0,0)--(0,1),W);
 +
</asy>
 +
We will now pick arbitrary points <math>E</math> and <math>F</math> on <math>\overline{AB}</math> and <math>\overline{AD}</math> respectively. We shall say that <math>AE = AF = x</math>
 +
<asy>
 +
draw((0,0)--(0,1));
 +
draw((0,1)--(1,1));
 +
draw((1,1)--(1,0));
 +
draw((1,0)--(0,0));
 +
label("A",(0,1),NW);
 +
label("B",(1,1),NE);
 +
label("C",(1,0),SE);
 +
label("D",(0,0),SW);
 +
label("$1$",(0,0)--(0,1),W);
 +
draw((1,0)--(0.5,1),blue);
 +
draw((0.5,1)--(0,0.5),blue);
 +
draw((0,0.5)--(0,0),blue);
 +
draw((0,0)--(1,0),blue);
 +
label("E",(0.5,1),N);
 +
label("F",(0,0.5),SE);
 +
</asy>
 +
Thus, our problem has been simplified to maximizing the area of the blue quadrilateral.
 +
If we drop an altitude from <math>E</math> to <math>\overline{DC}</math>, and call the foot of the altitude <math>G</math>, we can find the area of <math>EFDC</math> by noting that <math>[EFDC] = [EGDF] + [EGC]</math>.
 +
<asy>
 +
draw((0,0)--(0,1));
 +
draw((0,1)--(1,1));
 +
draw((1,1)--(1,0));
 +
draw((1,0)--(0,0));
 +
label("A",(0,1),NW);
 +
label("B",(1,1),NE);
 +
label("C",(1,0),SE);
 +
label("D",(0,0),SW);
 +
label("$1$",(0,0)--(0,1),W);
 +
draw((1,0)--(0.5,1),blue);
 +
draw((0.5,1)--(0,0.5),blue);
 +
draw((0,0.5)--(0,0),blue);
 +
draw((0,0)--(1,0),blue);
 +
label("E",(0.5,1),N);
 +
label("F",(0,0.5),SE);
 +
draw((0.5,1)--(0.5,0),red);
 +
draw((0.5,0.1)--(0.4,0.1),red);
 +
draw((0.4,0.1)--(0.4,0),red);
 +
label("G",(0.5,0),S);
 +
</asy>
 +
We can now finish the problem.
 +
 
 +
Since <math>EG = 1</math> and <math>FD = 1-x</math>, we have:
 +
<cmath>[EFDC] = [EGDF] + [EGC]
 +
= DG\cdot\frac{EG + FD}{2} + \frac{(CG)(EG)}{2}
 +
= \frac{x(2-x)}{2} + \frac{1-x}{2}
 +
= \frac{1+x-x^2}{2}</cmath>
 +
 
 +
To maximize this, we compete the square in the numerator to have:
 +
<cmath>[EFDC] = \frac{-(x-\frac{1}{2})^2 + \frac{5}{4}}{2}
 +
= \frac{5}{8} - \frac{(x-\frac{1}{2})^2}{2}</cmath>
 +
 
 +
Finally, we see that <math>\frac{(x-\frac{1}{2})^2}{2}\geq0</math>, as:
 +
<cmath>\left(x-\frac{1}{2}\right)^2\geq0</cmath>
 +
So,
 +
<cmath>\frac{(x-\frac{1}{2})^2}{2}\geq0,</cmath>
 +
 
 +
where the first inequality was from the Trivial Inequality, and the second came from dividing both sides by <math>2</math>, which does not change the inequality sign.
 +
Thus, the maximum area is <math>\boxed{\frac{5}{8}}</math> or <math>\boxed{\textbf{(D)}},</math> when <math>\frac{(x-\frac{1}{2})^2}{2} = 0.</math>
 +
 
 +
==Solution 3 (Calculus)==
 +
 
 +
Let <math>AE = AF = x</math>.
 +
 
 +
The area of the quadrilateral <math>CDFE</math> can be expressed as <math>A = 1 - \frac{x^2}{2} - \frac{(1-x)}{2}</math>.
 +
 
 +
By taking the derivative of <math>A</math>, we get <math>A' = -x + \frac{1}{2}</math>.
 +
 
 +
We make <math>A' = 0</math> and get the critical point <math>x = \frac{1}{2}</math>.
 +
 
 +
Substituting <math>x = \frac{1}{2}</math> , the maximum area is <math>\boxed{\frac{5}{8}}</math> or <math>\boxed{\textbf{(D)}}</math>.
 +
 
 +
~belanotbella
 +
 
 +
==Solution 4==
 +
Plot the points of the quadrilateral (with the bottom left corner of the square being at the origin) to plug into shoelace formula:
 +
<cmath>(0, 0), (0, 1-x), (x, 1), (1, 0)</cmath>
 +
So,
 +
<cmath>\frac{-x^2+x+1}{2}</cmath>
 +
Trying <math>1/2</math> first, you get <math>5/8</math>. Any larger or smaller value gives an answer less than <math>5/8</math>. Therefore, the answer is <math>\boxed{D}</math>.
 +
 
 +
~MC413551

Latest revision as of 08:08, 13 April 2023

Problem

$ABCD$ is a square with side of unit length. Points $E$ and $F$ are taken respectively on sides $AB$ and $AD$ so that $AE = AF$ and the quadrilateral $CDFE$ has maximum area. In square units this maximum area is:

$\textbf{(A)}\ \frac{1}2\qquad\textbf{(B)}\ \frac{9}{16}\qquad\textbf{(C)}\ \frac{19}{32}\qquad\textbf{(D)}\ \frac{5}{8}\qquad\textbf{(E)}\ \frac{2}3$

Solution

Let $AE=AF=x$ $[CDFE]=[ABCD]-[AEF]-[EBC]=1-\frac{x^2}{2}-\frac{1-x}{2}$ Or $[CDFE]=\frac{\frac{5}{4}-(x-\frac{1}{2})^2}{2}\le \frac{5}{8}$ As $(x-\frac{1}{2})^2\ge 0$ So $[CDFE]\le \frac{5}{8}$ Equality occurs when $AE=AF=x=\frac{1}{2}$ So the maximum value is $\frac{5}{8}$

Solution 2

Let us first draw a unit square. [asy] draw((0,0)--(0,1)); draw((0,1)--(1,1)); draw((1,1)--(1,0)); draw((1,0)--(0,0)); label("A",(0,1),NW); label("B",(1,1),NE); label("C",(1,0),SE); label("D",(0,0),SW); label("$1$",(0,0)--(0,1),W); [/asy] We will now pick arbitrary points $E$ and $F$ on $\overline{AB}$ and $\overline{AD}$ respectively. We shall say that $AE = AF = x$ [asy] draw((0,0)--(0,1)); draw((0,1)--(1,1)); draw((1,1)--(1,0)); draw((1,0)--(0,0)); label("A",(0,1),NW); label("B",(1,1),NE); label("C",(1,0),SE); label("D",(0,0),SW); label("$1$",(0,0)--(0,1),W); draw((1,0)--(0.5,1),blue); draw((0.5,1)--(0,0.5),blue); draw((0,0.5)--(0,0),blue); draw((0,0)--(1,0),blue); label("E",(0.5,1),N); label("F",(0,0.5),SE); [/asy] Thus, our problem has been simplified to maximizing the area of the blue quadrilateral. If we drop an altitude from $E$ to $\overline{DC}$, and call the foot of the altitude $G$, we can find the area of $EFDC$ by noting that $[EFDC] = [EGDF] + [EGC]$. [asy] draw((0,0)--(0,1)); draw((0,1)--(1,1)); draw((1,1)--(1,0)); draw((1,0)--(0,0)); label("A",(0,1),NW); label("B",(1,1),NE); label("C",(1,0),SE); label("D",(0,0),SW); label("$1$",(0,0)--(0,1),W); draw((1,0)--(0.5,1),blue); draw((0.5,1)--(0,0.5),blue); draw((0,0.5)--(0,0),blue); draw((0,0)--(1,0),blue); label("E",(0.5,1),N); label("F",(0,0.5),SE); draw((0.5,1)--(0.5,0),red); draw((0.5,0.1)--(0.4,0.1),red); draw((0.4,0.1)--(0.4,0),red); label("G",(0.5,0),S); [/asy] We can now finish the problem.

Since $EG = 1$ and $FD = 1-x$, we have: \[[EFDC] = [EGDF] + [EGC] = DG\cdot\frac{EG + FD}{2} + \frac{(CG)(EG)}{2} = \frac{x(2-x)}{2} + \frac{1-x}{2} = \frac{1+x-x^2}{2}\]

To maximize this, we compete the square in the numerator to have: \[[EFDC] = \frac{-(x-\frac{1}{2})^2 + \frac{5}{4}}{2} = \frac{5}{8} - \frac{(x-\frac{1}{2})^2}{2}\]

Finally, we see that $\frac{(x-\frac{1}{2})^2}{2}\geq0$, as: \[\left(x-\frac{1}{2}\right)^2\geq0\] So, \[\frac{(x-\frac{1}{2})^2}{2}\geq0,\]

where the first inequality was from the Trivial Inequality, and the second came from dividing both sides by $2$, which does not change the inequality sign. Thus, the maximum area is $\boxed{\frac{5}{8}}$ or $\boxed{\textbf{(D)}},$ when $\frac{(x-\frac{1}{2})^2}{2} = 0.$

Solution 3 (Calculus)

Let $AE = AF = x$.

The area of the quadrilateral $CDFE$ can be expressed as $A = 1 - \frac{x^2}{2} - \frac{(1-x)}{2}$.

By taking the derivative of $A$, we get $A' = -x + \frac{1}{2}$.

We make $A' = 0$ and get the critical point $x = \frac{1}{2}$.

Substituting $x = \frac{1}{2}$ , the maximum area is $\boxed{\frac{5}{8}}$ or $\boxed{\textbf{(D)}}$.

~belanotbella

Solution 4

Plot the points of the quadrilateral (with the bottom left corner of the square being at the origin) to plug into shoelace formula: \[(0, 0), (0, 1-x), (x, 1), (1, 0)\] So, \[\frac{-x^2+x+1}{2}\] Trying $1/2$ first, you get $5/8$. Any larger or smaller value gives an answer less than $5/8$. Therefore, the answer is $\boxed{D}$.

~MC413551

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