A ship P sails at an average speed of 50 km/h on a bearing of N35°E from a port R. At the same time another ship Q sails at an average speed of 40 km/h on a bearing of S25°W. How far are they after 4 hours?
All angles are measured CW from +y-axis.
RP = 50[35o] * 4 = 200km[35o].
PR = 200[35+180] = 200km[215o].
RQ = 40[205o] * 4 = 160km[205o].
PQ = PR+RQ = 200[215o] + 160[205o]
PQ = (200*sin215+160*sin205) + (200*cos215+160*cos205)I
PQ = -182. - 309i. = 359km[30.5o].
After 4 hours,
P has gone 200 km
Q has gone 160 km
The angle between their headings is 170°
Now just use the law of cosines to find the distance PQ
Well, it seems like ships P and Q are playing a game of hide and seek in the ocean!
Let's find out where they hid after 4 hours. Ship P sailed at an average speed of 50 km/h on a bearing of N35°E. This means it sailed in a northward direction, slightly to the east. Since it sailed for 4 hours at this speed, it covered a distance of 50 km/h * 4 h = 200 km.
On the other hand, ship Q sailed at an average speed of 40 km/h on a bearing of S25°W. That means it sailed in a southward direction, slightly to the west. So after 4 hours, it also covered a distance of 40 km/h * 4 h = 160 km.
Now, we need to figure out their distance apart. We can draw a right-angled triangle with one side being 200 km (P's distance) and the other side being 160 km (Q's distance). The hypotenuse of this triangle represents the distance between the two ships.
Using the Pythagorean theorem (or math clown magic), we can calculate the hypotenuse:
Distance^2 = 200 km^2 + 160 km^2
Distance^2 = 40000 km^2 + 25600 km^2
Distance^2 = 65600 km^2
Distance = sqrt(65600) km
Distance ≈ 256 km
So, after 4 hours, ships P and Q are approximately 256 km apart. I hope they're enjoying their game of hide and seek!
To find the distance between ships P and Q after 4 hours, we need to calculate their respective displacements.
For ship P:
Distance = Speed * Time
Distance = 50 km/h * 4 hours = 200 km
For ship Q:
Distance = Speed * Time
Distance = 40 km/h * 4 hours = 160 km
Now, we can use the law of cosines to calculate the distance between the two ships. The law of cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides, minus twice the product of the lengths of those two sides, multiplied by the cosine of the included angle.
Let's name the distance between the two ships as "d."
Using the law of cosines:
d^2 = 200^2 + 160^2 - 2 * 200 * 160 * cos(180° - 35° - 205°)
d^2 = 40,000 + 25,600 - 64,000 * cos(160°)
d^2 = 65,600 - 64,000 * -0.9396
d^2 = 125,600 - (-60,070.4)
d^2 = 185,670.4
d ≈ √(185,670.4)
d ≈ 431.1 km
Therefore, the ships are approximately 431.1 km apart after 4 hours.
To find the distance between the two ships after 4 hours, we need to calculate the distance traveled by each ship separately.
Let's start with Ship P:
Ship P sails at an average speed of 50 km/h on a bearing of N35°E. This means that Ship P is moving in a direction 35 degrees east of north.
To find the distance traveled by Ship P after 4 hours, we can use the formula: Distance = Speed × Time.
Distance_P = Speed_P × Time_P
Distance_P = 50 km/h × 4 h
Distance_P = 200 km
So, Ship P has traveled a distance of 200 km after 4 hours.
Now let's calculate the distance traveled by Ship Q:
Ship Q sails at an average speed of 40 km/h on a bearing of S25°W. This means that Ship Q is moving in a direction 25 degrees west of south.
To find the distance traveled by Ship Q after 4 hours, we can use the same formula: Distance = Speed × Time.
Distance_Q = Speed_Q × Time_Q
Distance_Q = 40 km/h × 4 h
Distance_Q = 160 km
So, Ship Q has traveled a distance of 160 km after 4 hours.
To find the distance between the two ships, we can use the Pythagorean theorem, as they have traveled in perpendicular directions. The Pythagorean theorem states that the square of the hypotenuse (c) of a right-angled triangle is equal to the sum of the squares of the other two sides (a and b).
In this case, the distances traveled by Ship P and Ship Q form the two sides of the right triangle, and the distance between the ships is the hypotenuse.
Distance^2 = Distance_P^2 + Distance_Q^2
Distance^2 = 200^2 + 160^2
Distance^2 = 40000 + 25600
Distance^2 = 65600
Distance = √65600
Distance ≈ 256.1 km
Therefore, after 4 hours, the two ships are approximately 256.1 km apart.