Find an objective function for total time that uses rowing speed as an independent​ variable, then use calculus to find the rowing speed that minimizes the function when a direct route to the restaurant is taken.

To solve this, let’s define an objective function that represents the **total time taken** to reach the restaurant by rowing across a river and walking a remaining distance. We’ll follow these steps:

1. Define variables for the distances, rowing and walking speeds.
2. Set up the objective function for total time using the rowing speed as the independent variable.
3. Differentiate and find the optimal rowing speed to minimize total time.

### Step 1: Define Variables and Assumptions

- Let \( v_r \) be the rowing speed (in units like mph or km/h).
- Let \( d_r \) be the direct rowing distance to the restaurant.
- Let \( v_w \) be the walking speed.
- Let \( d_w \) be the walking distance (if any after rowing).
  
The total time, \( T \), to reach the restaurant consists of rowing time and walking time:

\[
T = \frac{d_r}{v_r} + \frac{d_w}{v_w}
\]

### Step 2: Set Up Objective Function for Total Time

For a **direct route** to the restaurant, assume that you row directly to the point where you would also begin walking (if any walking is required), minimizing the time spent. So, the objective function \( T(v_r) \) with \( v_r \) as the independent variable becomes:

\[
T(v_r) = \frac{d_r}{v_r} + \frac{d_w}{v_w}
\]

### Step 3: Optimize Total Time with Calculus

Since \( T(v_r) \) is the total time function, we want to **minimize \( T(v_r) \)** with respect to \( v_r \). This involves:

1. Taking the derivative \( T'(v_r) \) with respect to \( v_r \).
2. Setting \( T'(v_r) = 0 \) to find the critical points.
3. Verifying the minimum by checking the second derivative or using other means (like analyzing the behavior of \( T(v_r) \)).

### Differentiating \( T(v_r) \) with Respect to \( v_r \)

\[
T(v_r) = \frac{d_r}{v_r} + \frac{d_w}{v_w}
\]

Calculating the derivative:

\[
T'(v_r) = -\frac{d_r}{v_r^2}
\]

Setting \( T'(v_r) = 0 \):

\[
-\frac{d_r}{v_r^2} = 0
\]

This suggests

Find an objective function for total time that uses rowing speed as an independent variable, then use calculus to find the rowing speed that minimizes the function when a direct route to the restaurant is taken.

Learn how to optimize rowing speed to minimize total travel time using calculus. This problem involves setting up an objective function for time as a function of rowing speed and applying derivatives to find the optimal solution. Suitable for advanced high school students or introductory calculus learners.

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