Question content area top
Part 1
A boat on the ocean is 4 mi from the nearest point on a straight​ shoreline; that point is 8 mi from a restaurant on the shore. A woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant. Complete parts​ (a) and​ (b) below.
4 mi
8
mi
Question content area bottom
Part 1
a.  If she walks at 3​ mi/hr and rows at 2​ mi/hr, at which point on the shore should she land to minimize the total travel​ time?
To minimize the total travel​ time, the boat should land 
  
enter your response here miles from the restaurant.
​(Type an exact​ answer, using radicals as​ needed.) 

To minimize the total travel time, we need to express the time it takes for the woman to row to a point on the shore and then walk to the restaurant in terms of where she lands on the shore, and then find the value that minimizes this total time.

### Problem Setup:
- The woman starts 4 miles away from the shore, and the restaurant is 8 miles along the shore from the nearest point on the shoreline to her current position.
- She rows to some point \( x \) miles from the nearest point on the shore to the restaurant, and then walks the remaining distance to the restaurant.

### Step 1: Expression for Rowing Time
Let the distance from the point on the shore where she lands to the nearest point on the shoreline be \( x \) miles. The distance she rows, which is the hypotenuse of a right triangle, is:
\[
\text{Rowing Distance} = \sqrt{4^2 + x^2} = \sqrt{16 + x^2}.
\]
Her rowing speed is 2 miles per hour, so the time to row this distance is:
\[
\text{Rowing Time} = \frac{\sqrt{16 + x^2}}{2}.
\]

### Step 2: Expression for Walking Time
Once she lands on the shore, she still needs to walk along the shoreline to the restaurant. The remaining distance to the restaurant is \( 8 - x \) miles. Her walking speed is 3 miles per hour, so the time to walk this distance is:
\[
\text{Walking Time} = \frac{8 - x}{3}.
\]

### Step 3: Total Travel Time
The total travel time \( T(x) \) is the sum of the rowing time and the walking time:
\[
T(x) = \frac{\sqrt{16 + x^2}}{2} + \frac{8 - x}{3}.
\]

### Step 4: Minimize Total Travel Time
To minimize the total travel time, we need to find the value of \( x \) that minimizes \( T(x) \). This requires taking the derivative of \( T(x) \) with respect to \( x \), setting it equal to 0, and solving for \( x \).

#### Derivative of \( T(x) \):
\[
T'(x) = \frac{1}{2} \cdot \frac{x}{\sqrt{16 + x^2}} - \frac{1}{3}.
\]
Set \( T'(x) = 0 \) to find the critical points:
\[
\frac{x}{2\sqrt{16 + x^2}} = \frac{1}{3}.
\]
Multiply both sides by \( 2\sqrt{16 + x^2} \) to eliminate the fraction:
\[
x = \frac{2}{3} \sqrt{16 + x^2}.
\]
Square both sides to get rid of the square root:
\[
x^2 = \frac{4}{9}(16 + x^2).
\]
Multiply both sides by 9 to simplify:
\[
9x^2 = 4(16 + x^2).
\]
Expand and simplify:
\[
9x^2 = 64 + 4x^2,
\]
\[
5x^2 = 64,
\]
\[
x^2 = \frac{64}{5},
\]
\[
x = \frac{8}{\sqrt{5}} = \frac{8\sqrt{5}}{5}.
\]

### Final Answer:
The boat should land at \( \frac{8\sqrt{5}}{5} \) miles from the nearest point on the shore, which is approximately \( 3.58 \) miles from the restaurant.

Would you like further details on any part of the process? 

Here are 5 questions to extend this topic:
1. How does the total travel time change if the rowing speed increases?
2. What happens to the optimal landing point if the restaurant's distance changes?
3. How can you apply this problem's method to optimize a different route scenario?
4. Can you use graphical methods to visualize the minimum travel time?
5. What if the woman could swim to the shore faster—how would that affect the optimal landing point?

**Tip:** In problems involving minimization, always express total quantities (like time) in terms of a single variable before differentiating.

A boat on the ocean is 4 mi from the nearest point on a straight shoreline; that point is 8 mi from a restaurant on the shore. A woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant. If she walks at 3 mi/hr and rows at 2 mi/hr, at which point on the shore should she land to minimize the total travel time?

Solve an optimization problem involving a boat 4 miles from shore and a restaurant 8 miles along the shore. Learn how to minimize travel time using calculus and geometry. Rowing speed is 2 mi/h, and walking speed is 3 mi/h. Suitable for advanced high school students (Grades 11-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation