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Previous Puzzles of the Month + Solutions

 
June-July 2007, Puzzle nr 112
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Puzzle # 112  Italiano italiano Français francais
Difficulty level: bulbbulbbulb, basic geometry knowledge.

Achtung, Minen!
   In the late 40’s a soldier named Mark entered accidentally a square mined field (point A, see drawing below)... Two friends of him, at the edge of the field, shouted him and asked to continue walking directly towards the shelter B made from sand bags...
   So, all together from different directions started going straight ahead to the shelter. The average walking speed of the three soldiers was habitually 6 km/h, but Mark while he was crossing the mined field walked carefully at an average speed of 3 km/h. The two friends, at the edge of the field, were separated from each other by approx. 1060 meters.
   Nevertheless, all three arrived simultaneously at the shelter 10 minutes later. Can you indicate on the drawing where initially did Mark’s friends stand and what is the area of the mined field? To solve this puzzle use only basic geometry, trigonometry is not allowed.

puzzle 112
[enlarge]


solution
This problem isn't actually very difficult... We know that the friends of Mark at the edges of the square field walked directly toward the shelter B and arrived at the same time; then, their initial position can be defined by points C and D where the perimeter of the square intersects the arc of a circle having center B. But we need just an extra information to find the exact places where Mark’s friends stand...

puzzle solution 1

• Draw a line through points A and B (see drawing below).
• Draw a circle having center A, whose radius r meets the intersection point between the side of the square field and the line AB. The diameter of this circle (2 x r) represents the distance Mark might have covered inside the field with a normal average speed (6 km/h = 2 x 3 km/h) for the same amount of time.
• Draw a large circle having center B and being tangent to the circle whose center is A. The arc of circle having center B determines now two points C and D on the perimeter of the square, which are the initial positions of Mark's friends.
• BC = BD, then triangle CDB is isosceles.

puzzle solution 2

CB and DB are the distances that Mark's friends effectively covered which are in meters: 10/60 x 6 [km/h] = 1 [km] = 1000 meters

Note that the height h of the triangle CDB, is also the height of the square field.

triangleIn any triangle it is possible to find the height h, if the value of its sides a, b and c are known, using the following formula which is a variant of Hero's formula:
h = 2square root(s(s-a)(s-b)(s-c))/b
where s is the semiperimeter of the triangle:
s = (a+b+c)/2

Now adapting the above formula to our triangle CDB, we obtain:
h = 2square root(s(s-b)(s-d)2)/d
where s = (2 x 1000 + 1060)/2 = 1530
then,
h = 1/500 x square root(1530 x (1530 - 1060) x (1530 - 1000)2) = 898.88 [m]

Thus, the Area A of the square field is: 898.882 = 807,980.76 [m2] or approximately 0.808 [km2]

Simple, isn't it?

 

cup winnerThe Winners of the Puzzle of the Month are:
Paris Karagounis, Greece greec flag
Thierry Lebordais, France french flag

Congratulations!

 

© 2003 G. Sarcone, www.archimedes-lab.org
You can re-use content from Archimedes’ Lab on the ONLY condition that you provide credit to the authors (© G. Sarcone and/or M.-J. Waeber) and a link back to our site. You CANNOT reproduce the content of this page for commercial purposes.

You're encouraged to expand and/or improve this article. Send your comments, feedback or suggestions to Gianni A. Sarcone. Thanks!



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