# Math Help - Differentiate

1. ## Differentiate

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

2. 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.