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**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.