Differentiate

Quote:

Originally Posted by tommylai12

A man in a boat is 3 km from O, which is the nearest point in straight beach. His destinations is 6 km along the beach from O. If he can row at 4km/h and walk at 5km/h towards what point on the beach should he row to reach his destination in the least possible time? Ans: 4km from O

Code:

o B |\ | \ | \ | \ | \ o-----o---------o beach O X D

B = boat
D = destination
OB = 3 km
OD = 8 km
Let OX = x km.
Time to reach X from B: $t_1(x)=\frac{\sqrt{BO^2 + OX^2}}{4}=\frac{\sqrt{9 + x^2}}{4}$ .
Time to reach D from X: $t_2(x)=\frac{XD}{5}=\frac{8-x}{5}$
Total time: $t(x)=t_1(x)+t_2(x)=\frac{\sqrt{9 + x^2}}{4}+\frac{8-x}{5}$ , where x varies from 0 to 8.

Differentiate $t(x)$ w.r.t. x, then solve $t'(x)=0$ . The answer should be x=4.