Sure, let's break down the problem step by step to determine the distance between the two ships, A and B, at 14:00.
### Step 1: Determine the time traveled
Both ships travel from 11:00 to 14:00, which is a total of 3 hours.
### Step 2: Calculate distances traveled by each ship
- Ship A travels at a speed of 25 km/h:
[tex]\[
\text{Distance traveled by A} = 25 \text{ km/h} \times 3 \text{ hours} = 75 \text{ km}
\][/tex]
- Ship B travels at a speed of 20 km/h:
[tex]\[
\text{Distance traveled by B} = 20 \text{ km/h} \times 3 \text{ hours} = 60 \text{ km}
\][/tex]
### Step 3: Convert bearings to coordinates
To find the position of each ship, we use trigonometry. Bearings are provided in degrees and need to be converted to coordinates.
For Ship A (bearing of [tex]\( 080^\circ \)[/tex]):
- x-coordinate of A:
[tex]\[
x_A = 75 \times \sin(80^\circ) \approx 73.86 \text{ km}
\][/tex]
- y-coordinate of A:
[tex]\[
y_A = 75 \times \cos(80^\circ) \approx 13.02 \text{ km}
\][/tex]
For Ship B (bearing of [tex]\( 152^\circ \)[/tex]):
- x-coordinate of B:
[tex]\[
x_B = 60 \times \sin(152^\circ) \approx 28.17 \text{ km}
\][/tex]
- y-coordinate of B:
[tex]\[
y_B = 60 \times \cos(152^\circ) \approx -52.98 \text{ km}
\][/tex]
### Step 4: Calculate the distance between the coordinates
Now, we use the distance formula between the coordinates [tex]\((x_A, y_A)\)[/tex] and [tex]\((x_B, y_B)\)[/tex]:
[tex]\[
\text{Distance between ships} = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}
\][/tex]
Substituting the calculated values:
[tex]\[
\text{Distance} = \sqrt{(28.17 - 73.86)^2 + (-52.98 - 13.02)^2}
\][/tex]
Solving this:
[tex]\[
\text{Distance} = \sqrt{(-45.69)^2 + (-66.00)^2}
\][/tex]
[tex]\[
\text{Distance} \approx \sqrt{2087.28 + 4356.00}
\][/tex]
[tex]\[
\text{Distance} \approx \sqrt{6443.28}
\][/tex]
[tex]\[
\text{Distance} \approx 80.27 \text{ km}
\][/tex]
Thus, the distance between ships A and B at 14:00 is approximately 80.27 km.
