Think and Reflect
Q1. In my school, the playground is too small to have a 400 m track, so the school constructed a 200 m track instead. Does this mean that we need a smaller stagger for the race tracks in my school (i.e., smaller than the stagger used in the Olympics), for the same 4 × 100 m relay race?
Answer:
Yes, a smaller stagger would be needed. The stagger compensates for the extra distance a runner in an outer lane covers on the curved portions of the track. If the track is shorter (200 m instead of 400 m), the radius of the curves is likely smaller. The difference in the circumference between lanes depends on the radius of the curves and the lane width. With smaller curves (smaller radius), the additional distance per lane is proportionally less, meaning the stagger required is smaller than for a 400 m track.
Q2. What is the connection between this question and the one about the 400 m athletics track?
Answer:
The question about the perimeter of a circle (circumference) is directly connected to the athletics track problem. The curved parts of the track are semicircles, and the stagger distance between lanes is calculated by finding the difference in the circumferences of the circles formed by those semicircles. Without knowing how to calculate the circumference (the perimeter of a circle), you cannot determine the stagger needed for the race.
Q1. What procedure would you use to square a given triangle? Here, the task is to construct a square whose area is equal to the area of some given triangle. Think carefully. How would you proceed?
Answer:
First, find the area of the given triangle using the formula (1/2) × base × height or Heron's formula. Let the area be A. Then, construct a square whose side length is √A. This can be done geometrically by constructing a right-angled triangle with hypotenuse A+1 and one side A-1, then the other side will be 2√A, or by using the geometric mean method. Alternatively, use the method of 'squaring a rectangle' by first converting the triangle into an equal-area rectangle (by taking base and half the height), then squaring that rectangle using Baudhāyana's construction.
Q2. Why were human beings so fond of using circular shapes? Was this only for practical reasons, or could there have been other reasons too? What kinds of uses have human beings found for the circular shape?
Answer:
Human beings used circular shapes for both practical and aesthetic reasons. Practically, circles are easy to draw with a rope and peg, wheels (circular) allow smooth motion, cylindrical containers hold maximum volume for a given amount of material, and circular shapes distribute stress evenly (arches, domes). Aesthetically, circles represent perfection, symmetry, and infinity in many cultures. Uses include: wheels, pots, buildings (circular huts, amphitheaters), coins, clocks, and celestial observations (sun, moon).