Question

One ship leaves a port and sails at 17km/h on a bearing of 024 degrees. A second ship leaves the same port at the same time and sails at 21km/h on a bearing of 079 degrees. How far apart are 2 ships after 2 hours.

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Neal

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58 Answers

Step 1: Find the displacement vector for each ship after 2 hours.

Letd_1 be the displacement vector for the first ship and d_2 be the displacement vector for the second ship.

The displacement vector is calculated using the formula:

\textbf{d} = \textbf{v} \times \textbf{t}

where:

\textbf{v} = velocity vector,

\textbf{t} = time vector.

For the first ship:

\textbf{v}_1 = 17 \, \text{km/h}

\textbf{t} = 2 \, \text{hours}

Calculating the displacement vector for the first ship:

d_1 = 17 \times 2 = 34 \, \text{km} \, (\text{bearing} \, 024^\circ)

For the second ship:

\textbf{v}_2 = 21 \, \text{km/h}

\textbf{t} = 2 \, \text{hours}

Calculating the displacement vector for the second ship:

d_2 = 21 \times 2 = 42 \, \text{km} \, (\text{bearing} \, 079^\circ)

Step 2: Find the distance between the two ships.

To find the distance between the two ships, we need to find the vector sum of the two displacement vectors:

\textbf{D} = \sqrt{(d_{1x} - d_{2x})^2 + (d_{1y} - d_{2y})^2}

where:

The x-component ofd_1 = 34 \cos(24^\circ) ,

The y-component ofd_1 = 34 \sin(24^\circ) ,

The x-component ofd_2 = 42 \cos(79^\circ) ,

The y-component ofd_2 = 42 \sin(79^\circ) .

Calculating the x and y components:

d_{1x} = 34 \cos(24^\circ)

d_{1y} = 34 \sin(24^\circ)

d_{2x} = 42 \cos(79^\circ)

d_{2y} = 42 \sin(79^\circ)

Step 3: Find the distance between the two ships using the formula.

Substitute the x and y components into the formula:

\textbf{D} = \sqrt{(34\cos(24^\circ) - 42\cos(79^\circ))^2 + (34\sin(24^\circ) - 42\sin(79^\circ))^2}

Step 4: Calculate the distance and find the final answer.

\textbf{D} = \sqrt{(34\cos(24^\circ) - 42\cos(79^\circ))^2 + (34\sin(24^\circ) - 42\sin(79^\circ))^2}

\textbf{D}=\sqrt{(23.05)^2+(-27.4)^2}

\textbf{D}=\sqrt{531.3+750.76}

\textbf{D}=\sqrt{1282.06}

\textbf{D}\approx35.8\text{ km}

Therefore, the two ships are approximately 35.8 km apart after 2 hours.

\boxed{35.8\text{ km}}

Let

The displacement vector is calculated using the formula:

where:

For the first ship:

Calculating the displacement vector for the first ship:

For the second ship:

Calculating the displacement vector for the second ship:

Step 2: Find the distance between the two ships.

To find the distance between the two ships, we need to find the vector sum of the two displacement vectors:

where:

The x-component of

The y-component of

The x-component of

The y-component of

Calculating the x and y components:

Step 3: Find the distance between the two ships using the formula.

Substitute the x and y components into the formula:

Step 4: Calculate the distance and find the final answer.

Therefore, the two ships are approximately 35.8 km apart after 2 hours.

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