Statistics and probability.

Problem 58: Fielding ( vv ) In a game of cricket, a fielder is perfectly placed to catch a ball. She watches the ball in flight and takes the catch just in front of her eye. The angle between the horizontal and her line of sight at a time t after the ball is struck is 0. Show that - (tan?) is constant during the flight. The next ball is also struck in the direction of the fielder but at a different velocity. In order to be perfectly placed to catch the ball, the fielder runs at constant speed towards the batsman. Assuming that the ground is horizontal, show that again -(tan) is constant during the flight.Problem 6: The regular Reuleaux heptagon ( v ) B G F D E The diagram shows a British 50 pence coin. The seven arcs AB, BC, .. ., FG, GA are of equal length and each arc is formed from the circle of radius a having its centre at the vertex diametri- cally opposite the mid-point of the arc. Show that the area of the face of the coin is J - 7tan JT 14Problem 55: Hank's gold mine ( vv ) Hank's Gold Mine has a very long vertical shaft of height . A light chain of length / passes over a small smooth light fixed pulley at the top of the shaft. To one end of the chain is attached a bucket A of negligible mass and to the other a bucket B of mass m. The system is used to raise ore from the mine as follows. When bucket A is at the top it is filled with mass 2m of water and bucket B is filled with mass Am of ore, where 0 0 for a s r s b, and g is the inverse of f. (i) By making a suitable change of variable, prove that ( ((r)dr = b8 - an - g(y) dy . (1) where a = f(a) and # = f(6). Interpret this formula geometrically, by means of a sketch, in the case where a and a are both positive. Verify the result (O) for f (x) = et, a = 0, b = 1. (ii) Prove similarly and interpret the formula 2n / if (x)dx = m(b-8 - ato) - * [=() ]' dy. (2)