a boat sails due North at a speed of 120km/h and a wind blow at a speed of 40km/h due East .Find the actual speed of boat .use √10=3.1

To find the actual speed of the boat considering both its own speed and the wind speed, we need to calculate the resultant velocity using vector addition.

The boat's velocity vector \( \vec{V}_b \) is 120 km/h due North, and the wind's velocity vector \( \vec{V}_w \) is 40 km/h due East.

Using the Pythagorean theorem to find the magnitude of the resultant velocity \( \vec{V}_r \):

\[
|\vec{V}_r| = \sqrt{|\vec{V}_b|^2 + |\vec{V}_w|^2}
\]

Substituting the given values:

\[
|\vec{V}_r| = \sqrt{120^2 + 40^2}
\]

Calculate each term inside the square root:

\[
120^2 = 14400
\]
\[
40^2 = 1600
\]

Now, sum these values:

\[
|\vec{V}_r| = \sqrt{14400 + 1600} = \sqrt{16000}
\]

Using the approximation \( \sqrt{10} = 3.1 \):

\[
\sqrt{16000} = \sqrt{16 \times 1000} = 4 \times \sqrt{1000} = 4 \times 10 \times \sqrt{10} = 40 \times 3.1 = 124
\]

Therefore, the actual speed of the boat is approximately 124 km/h.

Do you want more details or have any questions about this problem? Here are 5 related questions you might find interesting:

1. How do you calculate the direction of the resultant velocity vector?
2. What is vector addition, and how is it used in physics?
3. How do you find the components of a vector in a coordinate system?
4. Can you explain the Pythagorean theorem in more detail?
5. How does wind affect the motion of a boat in real-world scenarios?

**Tip:** When dealing with vectors, always consider both magnitude and direction to find the resultant vector accurately.

Learn how to calculate the actual speed of a boat using vector addition and the Pythagorean Theorem. This problem involves determining the resultant velocity of a boat moving north and wind blowing east. Suitable for Grades 9-12.

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