O A and O B are two roads enclosing an angle of 120^ . X and Y start from ' O ' at the same time. X travels along O A with a speed of 4 ~km / h and Y travels along O B with a speed of 3 ~km / h . The rate at which the shortest distance between X and Y is increasing after 1 ~h is

Step-by-step solution

Given, speed of x is 4 ~km / h and y is 3 ~km / h . After time t the distance covered by x is 4 t and y is 3 t . Let shortest distance between x and y=A . Then by cosine law A²=(4 t)²+(3 t)²-(4 t)(3 t) 2 120^ A²=16 t²+9 t²-24 t² (- 1 2 ) A²=25 t²+12 t² A²=37 t² A= 37 t If t=1 ~h , then A= 37 ~km Now, differentiating Eq. (i) w.r.t. t , we get 2 A A^ =37(2 t) After t=1 ~h , we get 2 37 A^ =2(37) After t=1 ~h , we get A^ = 37 Thus, rate at which shortest distance A changes with time is 37 ~km / h .