Question

HELP NEEDED!
"Minimizing the Total Time"
Suk wants to go from point A to point B, as shown in the figure. Point B is 1 mile upstream and on the north side of the 0.4-mi-wide river. The speed of the current is 0.5 mph. Her boat will travel 4 mph in still water. When she gets to the north side she will travel by bicycle at 6 mph. Let (alpha) represent the bearing of her course and (beta) represent the drift angle, as shown in the figure.
D) Find the approximate angle (alpha) that minimizes the total time for the trip and find the corresponding (beta)
D) answers: 31.9 and 6.1

Answer 1

what figure

😱

Answer 2

The solution is a bit involved
First I solved for \( z= \alpha+\beta\)

the velocity across the river in terms of < horizontal, vertical> (or <east, north> if you like) is
< 4 sin z - 0.5, 4 cos z>
the distance across the river "vertically" is 0.4
so using rate * time = distance
4 t1 cos z = 0.4
\[ t_1= \frac{1}{10 \cos z }= \frac{\sec z}{10} \]

Answer 3

in that time we move upstream, horizontally a distance
\[ x = t_1 \cdot ( 4 \sin z - 0.5) = 0.4 \tan z - 0.05 \sec z \]

and the trip on the bike is distance 1-x
and takes time
\[ t_2= \frac{1-x}{6} = \frac{1 - ( 0.4 \tan z - 0.05 \sec z )}{6}\]

and the total trip takes time
\[ t_1+t_2= 0.1 \sec x + \frac{1}{6} - \frac{1}{15} \tan z +\frac{1}{120} \sec z\\
T= \frac{13}{120} \sec x - \frac{1}{15} \tan z + \frac{1}{6}
\]

Answer 4

take the derivative with respect to z, i.e. find
\[ \frac{dT}{dz} \]
set = 0, and solve for z

Answer 5

once we have z, we can find \(t_1\)
and then x
and then \[ \beta = \tan^{-1} \frac{x}{0.4} \]
and \(\alpha= z - \beta\)

Answer 6

Thanks