Take a bow

Geometry Level 5

You're lost amidst the jungle. Your bow has broken when you fell a ravine. The only thing you carry with you is a pocket knife, a tape-measure and some arrows in a bag. It's darkening and you can feel a chill in your spine, since fierce animals had started to roam by their habitat.

You climb a tree, find some strong bindweed and cogitate if can make a bow to your arrows.

Suppose you had found bindweed enough to weave a rope of \( 50 cm \). You're a skilled archer and know that for maximum precision of an arrow shot the arrow must have \( \frac {2} {3} \) of the bow length and that distance between the middle point of the bow and the middle point of the rope must be \( \frac {1} {3} \) of the arrow length. Your arrows have \( 25 cm \).

What must be the opening (in degrees) of the bow? Give your answer to 2 decimal places.

Details and assumptions

  • A bow is intended to be a circular section, which points have all the same distances from a single point, the centre of a circumference. Radius is the distance between this single point and any point of the bow;

  • If you need some trigonometrical functions, \( \sin (53.13ยบ)=0.8 \)


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