Difference between revisions of "1997 PMWC Problems/Problem I11"

Latest revision as of 11:44, 13 August 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 $6750\text{ cm}^2$ .

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

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

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

@@ Line 3: / Line 3: @@
 <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==

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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Difference between revisions of "1997 PMWC Problems/Problem I11"

Latest revision as of 11:44, 13 August 2014

Problem

Solution

See Also