# Differentiate

• Mar 24th 2009, 06:18 AM
tommylai12
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
• Mar 25th 2009, 12:35 AM
Ivan
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: $\displaystyle t_1(x)=\frac{\sqrt{BO^2 + OX^2}}{4}=\frac{\sqrt{9 + x^2}}{4}$.
Time to reach D from X: $\displaystyle t_2(x)=\frac{XD}{5}=\frac{8-x}{5}$
Total time: $\displaystyle t(x)=t_1(x)+t_2(x)=\frac{\sqrt{9 + x^2}}{4}+\frac{8-x}{5}$, where x varies from 0 to 8.

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