https://file.aiquickdraw.com/mathbot/file/1728542703240renacncb.jpg

To solve this problem, we need to determine how many times the two particles have the same velocity during the interval \( 0 \leq t \leq 10 \).

### Step 1: Find the velocity functions
The velocity is the derivative of the position function with respect to time. The position functions are:
- \( x_1(t) = \sin t \)
- \( x_2(t) = e^{-2t} - 1 \)

#### Velocity of particle 1:
\[
v_1(t) = \frac{d}{dt} [\sin t] = \cos t
\]

#### Velocity of particle 2:
\[
v_2(t) = \frac{d}{dt} [e^{-2t} - 1] = -2e^{-2t}
\]

### Step 2: Set the velocities equal to each other
We want to find when \( v_1(t) = v_2(t) \), so we set:
\[
\cos t = -2e^{-2t}
\]

### Step 3: Solve the equation
We now solve \( \cos t = -2e^{-2t} \). 

This is a transcendental equation, which means it cannot be solved algebraically. However, we can analyze it graphically or numerically to find the number of solutions in the interval \( 0 \leq t \leq 10 \).

### Step 4: Analyze graphically
Graphing both \( \cos t \) and \( -2e^{-2t} \) in the interval \( [0, 10] \), we observe that the two curves intersect at two points.

### Conclusion:
The two particles have the same velocity **two** times within the given interval. The correct answer is:

**(c) two**

---

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you solve transcendental equations graphically?
2. What methods can be used to approximate solutions for transcendental equations?
3. How would the solution change if the time interval was different?
4. Can the velocities of the particles ever be equal outside of this interval?
5. What are the implications of having equal velocities at different times?

**Tip:** When solving equations involving both trigonometric and exponential terms, graphical or numerical methods often provide the most practical solutions.

Two particles start at the origin and move along the x-axis. For 0 ≤ t ≤ 10, their respective position functions are given by x1 = sin t and x2 = e^(-2t) - 1. For how many values of t do the particles have the same velocity? (a) none (b) one (c) two (d) three (e) four

Determine how many times two particles have the same velocity in the interval 0 ≤ t ≤ 10, given their respective position functions. The problem involves solving the transcendental equation cos t = -2e^(-2t) using graphical or numerical methods. Suitable for advanced 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