During a dull football game,1/4 spectators were left after 1 hour. during the next hour 20,000 spectators left. there were 1/12 of the original number of spectators still watching the game. how many spectators were originally present.

Sum is 17 and their difference is 3

Prove (a) cosh2(x) − sinh2(x) = 1 and (b) 1 − tanh 2(x) = sech 2(x). solution (a) cosh2(x) − sinh2(x) = ex + e−x 2 2 − 2 = e2x + 2 + e−2x 4 − = 4 = . (b) we start with the identity proved in part (a): cosh2(x) − sinh2(x) = 1. if we divide both sides by cosh2(x), we get 1 − sinh2(x) cosh2(x) = 1 or 1 − tanh 2(x) = .

The distance of a chord from the centre of a circle is 12cm and the lenght of the chord is 10cm. find the radius of the circle