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Difference between revisions of "1997 PMWC Problems/Problem I11"
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==Problem==
 
==Problem==
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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 <math>6750 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>.  
−
[[Image:ABCD.gif]]
+
 
 +
<asy>
 +
/* File unicodetex not found. */
 +
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
 +
import graph; size(3.45cm);
 +
real labelscalefactor = 0.5; /* changes label-to-point distance */
 +
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
 +
pen dotstyle = black; /* point style */
 +
real xmin = -19.75, xmax = 39.09, ymin = -10.43, ymax = 20.84; /* image dimensions */
 +
/* draw figures */
 +
draw((0,3.45)--(0,0));
 +
draw((0,0)--(4.29,0));
 +
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==

Revision as of 14:33, 20 April 2014

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] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(3.45cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -19.75, xmax = 39.09, ymin = -10.43, ymax = 20.84; /* image dimensions */ /* draw figures */ draw((0,3.45)--(0,0)); draw((0,0)--(4.29,0)); 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

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

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

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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