Two people start from the same point. One walks east at 2 mi/h and the other walks northeast at 4 mi/h. How fast (in mi/h) is the distance between the people changing after 15 minutes? (Round your answer to three decimal places.)


This problem involves related rates and can be solved using the Pythagorean theorem because the motion forms a right triangle.

Let:
- \( x \) be the distance traveled east by the first person,
- \( y \) be the distance traveled northeast by the second person.

We need to find how fast the distance between the two people is changing after 15 minutes, or \( \frac{dz}{dt} \), where \( z \) is the distance between them.

### Step 1: Set up the coordinates and equations
- The first person is walking east at 2 mi/h, so after \( t \) hours, the distance traveled east is \( x = 2t \).
- The second person is walking northeast at 4 mi/h. Since northeast forms a 45-degree angle, the horizontal and vertical components of their distance are \( y_x = 4t \cos(45^\circ) \) and \( y_y = 4t \sin(45^\circ) \). Both of these components are equal because \( \cos(45^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2} \). Therefore, the distance traveled northeast is \( y = 4t \).

Now, by the Pythagorean theorem, the distance \( z \) between the two people is:
\[
z^2 = x^2 + y^2
\]
Substitute the expressions for \( x \) and \( y \):
\[
z^2 = (2t)^2 + (4t)^2
\]
\[
z^2 = 4t^2 + 16t^2 = 20t^2
\]
So,
\[
z = \sqrt{20}t = 2\sqrt{5}t
\]

### Step 2: Differentiate with respect to time \( t \)
Differentiate both sides of the equation \( z = 2\sqrt{5}t \) with respect to \( t \) to find \( \frac{dz}{dt} \):
\[
\frac{dz}{dt} = 2\sqrt{5}
\]

### Step 3: Evaluate at \( t = \frac{1}{4} \) (since 15 minutes = 1/4 hours)
\[
\frac{dz}{dt} = 2\sqrt{5} \approx 2 \times 2.236 = 4.472 \text{ mi/h}
\]

Thus, the rate at which the distance between the two people is changing after 15 minutes is approximately \( \boxed{4.472} \) mi/h.

Would you like further details on any of the steps or have any questions?

---

Here are 5 related questions to extend the discussion:
1. What happens if one person moves southwest instead of northeast?
2. How would the answer change if the speeds were doubled?
3. What is the general form of the related rates equation for two objects moving at different angles?
4. How would you handle this problem if they moved in opposite directions (north and south)?
5. What if the time interval changes to 30 minutes?

**Tip:** When solving related rates problems, always start by identifying which quantities are changing and write them in terms of time.

Two people start from the same point. One walks east at 2 mi/h and the other walks northeast at 4 mi/h. How fast (in mi/h) is the distance between the people changing after 15 minutes? (Round your answer to three decimal places.)

Learn how to solve a related rates problem involving two people walking at different speeds and angles. This example uses the Pythagorean theorem and trigonometry to calculate how fast the distance between them changes after 15 minutes. Suitable for high school calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation