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1997 PMWC Problems/Problem I11: Difference between revisions

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==Problem==
==Problem==
A rectangle <math>ABCD</math> is made up of five small congruent rectangles as shown in the given figure. Find the perimeter, in cm, of <math>ABCD</math> if its area is <math>6750\text{cm}^2</math>.  
A rectangle <math>ABCD</math> is made up of five small congruent rectangles as shown in the given figure. Find the perimeter, in cm, of <math>ABCD</math> if its area is <math>6750\text{ cm}^2</math>.  


<asy>
<asy>
/* File unicodetex not found. */
import cse5;
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import olympiad;
import graph; size(3.45cm);  
size(4cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pathpen=black;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pair A=(0,0),B=(0,-2.5),C=(3,-2.5),D=(3,0);
pen dotstyle = black; /* point style */
D(MP("A",A,W)--MP("B",B,W)--MP("C",C,E)--MP("D",D,E)--cycle);
real xmin = -19.75, xmax = 39.09, ymin = -10.43, ymax = 20.84; /* image dimensions */
D((0,-1.5)--(3,-1.5));
/* draw figures */
D((1,0)--foot((1,0),(0,-1.5),(3,-1.5)));
draw((0,3.45)--(0,0));  
D((2,0)--foot((2,0),(0,-1.5),(3,-1.5)));
draw((0,0)--(4.29,0));
D((1.5,-1.5)--(1.5,-2.5));</asy>
draw((0,3.45)--(4.29,3.45));
draw((4.29,3.45)--(4.29,0));  
draw((0,1.32)--(4.29,1.32));  
draw((2.14,0)--(2.14,1.32));
draw((1.43,1.32)--(1.43,3.45));  
draw((2.86,1.32)--(2.86,3.45));
/* dots and labels */
label("$B$", (-0.2,0), SW * labelscalefactor);
label("$A$", (-0.2,3.45), NW * labelscalefactor);
label("$C$", (4.29,0), SE * labelscalefactor);  
label("$D$", (4.29,3.45), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);  
/* end of picture */
//Credit to dasobson for the diagram</asy>


==Solution==
==Solution==
Line 45: Line 31:


<math>2(w+l)+6w=330</math>
<math>2(w+l)+6w=330</math>
==Solution 2==
We can label the short side of the smaller rectangles <math>a</math> and the long side <math>b</math>. Additionally, since all the rectangles are congruent we can say that the area of one of the rectangles is <math>\frac{6750}{5}=1350</math>. Using our labels we can get the equations for our systems, <math>ab=1350 \text{and} (a+b)(2b)=6750</math>.
<math>2ab+2b^2=6750 \text{ and } 2ab=2700</math>
Using elimination,
<math>2b^2=4050</math>
<math>b^2=2025 \implies b=45</math> which also means <math>a=30</math>.
We want to find the area of the big rectangle which is <math>2(a+b+2b)</math>
Plugging in our values we get:
<math>2(75+90)</math>
<math>2(165)</math>
<math>\text{Perimeter}=\boxed{330}</math>
[https://aops.com/wiki/index.php/User:Am24 ~AM24]


==See Also==
==See Also==

Latest revision as of 19:23, 5 September 2025

Problem

A rectangle $ABCD$ is made up of five small congruent rectangles as shown in the given figure. Find the perimeter, in cm, of $ABCD$ if its area is $6750\text{ cm}^2$.

[asy] import cse5; import olympiad; size(4cm); pathpen=black; pair A=(0,0),B=(0,-2.5),C=(3,-2.5),D=(3,0); D(MP("A",A,W)--MP("B",B,W)--MP("C",C,E)--MP("D",D,E)--cycle); D((0,-1.5)--(3,-1.5)); D((1,0)--foot((1,0),(0,-1.5),(3,-1.5))); D((2,0)--foot((2,0),(0,-1.5),(3,-1.5))); D((1.5,-1.5)--(1.5,-2.5));[/asy]

Solution

Let $l$ and $w$ be the length, and width, respectively, of one of the small rectangles.

$3w=2l$

$l=\dfrac{3}{2}w$

$6750= 5lw = \dfrac{15}{2}w^2$

$w=30$

$l=45$

The perimeter of the big rectangle is

$2(w+l)+6w=330$

Solution 2

We can label the short side of the smaller rectangles $a$ and the long side $b$. Additionally, since all the rectangles are congruent we can say that the area of one of the rectangles is $\frac{6750}{5}=1350$. Using our labels we can get the equations for our systems, $ab=1350 \text{and} (a+b)(2b)=6750$.

$2ab+2b^2=6750 \text{ and } 2ab=2700$

Using elimination,

$2b^2=4050$

$b^2=2025 \implies b=45$ which also means $a=30$.

We want to find the area of the big rectangle which is $2(a+b+2b)$

Plugging in our values we get:

$2(75+90)$

$2(165)$

$\text{Perimeter}=\boxed{330}$

~AM24

See Also

1997 PMWC (Problems)
Preceded by
Problem I10 		Followed by
Problem I12
I: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
T: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
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