Difference between revisions of "1997 PMWC Problems/Problem I11"
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| − | + | ==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>. | ||
| + | |||
| + | <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 <math>l</math> and <math>w</math> be the length, and width, respectively, of one of the small rectangles. | ||
| + | |||
| + | <math>3w=2l</math> | ||
| + | |||
| + | <math>l=\dfrac{3}{2}w</math> | ||
| + | |||
| + | <math>6750= 5lw = \dfrac{15}{2}w^2</math> | ||
| + | |||
| + | <math>w=30</math> | ||
| + | |||
| + | <math>l=45</math> | ||
| + | |||
| + | The perimeter of the big rectangle is | ||
| + | |||
| + | <math>2(w+l)+6w=330</math> | ||
| + | |||
| + | ==See Also== | ||
| + | {{PMWC box|year=1997|num-b=I10|num-a=I12}} | ||
| + | |||
| + | [[Category:Introductory Geometry Problems]] | ||
Latest revision as of 11:44, 13 August 2014
Problem
A rectangle 
is made up of five small congruent rectangles as shown in the given figure. Find the perimeter, in cm, of if its area is 
.
![[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]](http://latex.artofproblemsolving.com/c/c/2/cc22833656d1d99c75ba2eec1c839acb97a7d95c.png)
Solution
Let 
and 
be the length, and width, respectively, of one of the small rectangles.
The perimeter of the big rectangle is
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 | ||
