Problem Breakdown

Let’s analyze each part of the problem systematically.

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Part 1 (a): Minimize Total Travel Time

Let:

• $x$: Distance from the landing point on the shore to the restaurant sympy import symbols, sqrt, diff, solve

Define variables

x = symbols('x', positive=True) # Distance from the landing point to the restaurant d_boat_to_shore = 3 # Distance from boat to shore d_shore_to_restaurant = 13 # Total distance along the shore

Speeds

v_row = 2 # Rowing speed (mi/hr) v_walk = 3 # Walking speed (mi/hr)

Time equations

time_row = sqrt(d_boat_to_shore2 + x2) / v_row # Time to row to landing point time_walk = (d_shore_to_restaurant - x) / v_walk # Time to walk to the restaurant

Total time

total_time = time_row + time_walk

Minimize total time

time_derivative = diff(total_time, x) # Derivative of time w.r.t. x optimal_x = solve(time_derivative, x) # Solve for critical points

optimal_xThe optimal landing point on the shore is approximately:

$x = \frac{6\sqrt{5}}{5} \, \text{miles from the restaurant.}$

This is the point where the total travel time is minimized.

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Part 1 (b): Minimum Speed to Row Directly

Here, the woman rows directly without walking. We need to find the rowing speed $v_{row\_min}$ such that rowing directly is faster than the combined time of rowing and walking for any landing point.

Let’s calculate this: Direct rowing distance direct_distance = sqrt(d_boat_to_shore2 + d_shore_to_restaurant2) # Hypotenuse

Time to row directly

time_row_direct = direct_distance / v_row

Set condition: rowing directly must be equal or faster than combined time

v_row_min = symbols('v_row_min', positive=True) time_row_direct_new = direct_distance / v_row_min row_walk_time_combined = total_time.subs(x, optimal_x[0]) # Combined time at optimal point

Solve for minimum rowing speed

min_rowing_speed = solve(time_row_direct_new - row_walk_time_combined, v_row_min) min_rowing_speedThe minimum speed at which she must row directly (without walking) to ensure the quickest way to the restaurant is:

$v_{\text{row}} = -\frac{18\sqrt{890}}{631} + \frac{156\sqrt{178}}{631} \, \text{mi/hr.}$

This is the exact expression for the required speed. If you'd like, I can calculate an approximate numerical value for better clarity. Let me know!

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Do you want further details or have any questions?

Relative Questions:

1. What is the numerical approximation of the optimal landing point distance?
2. How do walking and rowing speeds affect the optimal point?
3. What would happen if the walking speed were slower than the rowing speed?
4. How can the optimal solution be graphed to visualize travel times?
5. What practical considerations might alter the theoretical optimal solution?

Tip: When solving optimization problems, always verify critical points and boundary conditions to ensure you find the global minimum.