https://file.aiquickdraw.com/mathbot/file/1727407316393w56cy0ox.jpg

To explain why the answer is G (5.34 × 10¹ s) for the problem about the boat crossing the river, let's break down the scenario and calculations step by step.

### Problem Recap:
- The width of the river is 246.0 m.
- The boat’s speed in still water is 5.50 m/s.
- The river current flows at 3.00 m/s.

The problem asks to calculate **the time required for the boat to reach the opposite shore** directly across from its starting point.

### Key Concept:
To reach the opposite shore directly across from the starting point, the boat must be directed at an angle that compensates for the river's current. The boat's speed in still water (5.50 m/s) is directed diagonally, but only the **component perpendicular to the river banks** will help the boat cross the river. This component of the velocity is needed to determine the time taken.

#### Step 1: Calculate the Perpendicular Component of the Boat’s Speed
The river flows at 3.00 m/s, which affects the boat's horizontal (parallel to riverbanks) velocity. However, the boat's actual crossing speed is **the perpendicular component of its velocity**.

To find the boat’s speed across the river, we use the Pythagorean theorem:

\[
v_{\text{cross}} = \sqrt{v_{\text{boat}}^2 - v_{\text{current}}^2}
\]

Substitute the values:
\[
v_{\text{cross}} = \sqrt{(5.50 \, \text{m/s})^2 - (3.00 \, \text{m/s})^2}
\]

\[
v_{\text{cross}} = \sqrt{30.25 - 9.00} = \sqrt{21.25}
\]

\[
v_{\text{cross}} \approx 4.61 \, \text{m/s}
\]

#### Step 2: Calculate the Time to Cross the River
Now, using the formula for time:
\[
t = \frac{d}{v}
\]
where:
- \( d = 246.0 \, \text{m} \) (the width of the river),
- \( v = 4.61 \, \text{m/s} \) (the perpendicular velocity).

Substitute the values:
\[
t = \frac{246.0}{4.61} \approx 53.4 \, \text{seconds}
\]

In scientific notation, this is:
\[
t \approx 5.34 \times 10^1 \, \text{seconds}
\]

### Conclusion:
The time for the boat to cross the river is approximately **53.4 seconds**, which corresponds to option **G (5.34 × 10¹ s)**.

---

Would you like more details or have any further questions? Here are some related questions to deepen your understanding:

1. How would the time change if the current increased?
2. What angle should the boat steer at to ensure it crosses directly?
3. How is the resultant velocity of the boat calculated when considering both boat speed and current?
4. If the boat's speed were slower, what would happen to the time it takes to cross?
5. What happens if the boat moves faster than the current?

**Tip:** Always resolve velocity vectors when objects move in two directions at once (e.g., boat speed vs. current).

Explain why the answer is G for the problem: A boat crossing a 246.0 m wide river with a speed of 5.50 m/s in still water, while the river flows at 3.00 m/s.

Find out how to calculate the time required for a boat to cross a 246.0 m wide river. The boat moves at 5.50 m/s in still water while the river current is 3.00 m/s. Learn how to decompose velocities using the Pythagorean theorem to find the crossing time.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation