Further applications of differentiation 1

The diagram shows a river,500m wide between the straight parallel banks where AD = 500m and   DB = 2000. A man swims across the river from A to C at 0.5 m/s and then he runs along the banks from C to B at 0.8 m/s

(a)If he swims in the direction AC making an angle of Ɵ and reaches C in t seconds, show that t = 1000cosec Ɵ

(b)If T(in seconds) is the total time taken from A to B, express T in terms of Ɵ

(c)Find the minimum value of T in minutes and seconds, correct to the nearest second.



Using trigo ratio,

sin Ɵ = 500/AC

AC = 500/sin Ɵ

 = 500cosec Ɵ

 0.5 = 500cosec Ɵ/t

 T = 1000cosec Ɵ



BC = 2000 – DC

BC = 2000 – 500/tan Ɵ

TBC = BC/s = 2500 – 625/tan Ɵ

Total time, T = tAC + tBC

= 1000cosec Ɵ + 2500 – 625/tan Ɵ