Two ships, A and B st out from port O simultaneously. A travels due north at 16 km/h, B due east at 13 km/h. The ships radio have a rang of 120. For how long would they remain in contact if B had travelled north-east
Range,r = 120 km
and since the two ships are travelling at 90° to each other, we use Pythagoras:
distance, d = sqrt(Va^2+Vb^2)*t
t=time in hours
Va = speed of ship A
Vb = speed of ship B
In other words, the ships are in range as long as
d<r,
t√(Va²+Vb²)< 120
t<120/(√(Va²+Vb²))
To find out how long the ships A and B would remain in contact, we first need to determine the distance between them when they are in contact.
Since ship A is traveling due north at 16 km/h and ship B is traveling due east at 13 km/h, we can use the Pythagorean theorem to find out the distance between them.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, ship A's trajectory is the north side, ship B's trajectory is the east side, and the distance between them is the hypotenuse.
Using the Pythagorean theorem, we have:
Distance^2 = (side A)^2 + (side B)^2
Distance^2 = (16 km/h)^2 + (13 km/h)^2
Distance^2 = 256 km^2/h^2 + 169 km^2/h^2
Distance^2 = 425 km^2/h^2
Taking the square root of both sides, we get:
Distance = √(425 km^2/h^2)
Distance ≈ 20.62 km
So, when ships A and B are in contact, the distance between them is approximately 20.62 km.
Now, let's consider that ship B had traveled northeast instead of due east. Traveling northeast means that ship B is moving at a 45-degree angle between north and east.
If the range of the ship's radio is 120 km, we can draw a circle centered at ship A with a radius of 120 km. The point where this circle intersects the trajectory of ship B's northeast path will determine how long the ships remain in contact.
To find this point, we can use the concept of similar triangles. Since the ship is traveling at a 45-degree angle, the distance it travels in the north and east directions will be equal.
Let's call the time they remain in contact as 't' hours.
In that time, ship B would have traveled 13 km/h * t hours northeast, so it would have covered 13t km in each direction.
Using the Pythagorean theorem again, the distance between the ships when the radio's range intersects the ship B's path is:
Distance^2 = (side A)^2 + (side B)^2
(120 km)^2 = (16 km/h * t)^2 + (13 km/h * t)^2
14400 km^2 = 256 km^2/h^2 * t^2 + 169 km^2/h^2 * t^2
14400 km^2 = (256 + 169) km^2/h^2 * t^2
14400 km^2 = 425 km^2/h^2 * t^2
Dividing by 425 km^2/h^2, we get:
t^2 = 14400 km^2 / (425 km^2/h^2)
t^2 ≈ 33.88 h^2
Taking the square root of both sides, we get:
t ≈ √(33.88 h^2)
t ≈ 5.82 hours
Therefore, they would remain in contact for approximately 5.82 hours if ship B had traveled northeast.