Question

At noon, ship A is 170 km west of ship B. Ship A is sailing east at 40 km/h and ship B is sailing north at 25 km/h. How fast (in km/hr) is the distance between the ships chanaina at 4:00 p.m.?

Answer 1

To find the rate at which the distance between the ships is changing, we can use the concept of relative velocity. The ships are moving in perpendicular directions, so we can treat their velocities as the legs of a right triangle. The rate at which the distance between the ships is changing at 4:00 p.m. is equal to the magnitude of the total velocity vector, which is the square root of the sum of the squares of the individual velocities.

Step-by-step explanation:

To find the rate at which the distance between the ships is changing, we can use the concept of relative velocity. The ships are moving in perpendicular directions, so we can treat their velocities as the legs of a right triangle. Ship A is moving east at 40 km/h and Ship B is moving north at 25 km/h. Using the Pythagorean theorem, we can find the magnitude of the total velocity vector.

Let's call the distance between the ships at noon 'd'. The velocity of Ship A is 40 km/h and the velocity of Ship B is 25 km/h. By using the Pythagorean theorem, we can find the magnitude of the total velocity vector:

v = sqrt((40 km/h)^2 + (25 km/h)^2)

Therefore, the rate at which the distance between the ships is changing at 4:00 p.m. is equal to the magnitude of the total velocity vector, which is the square root of the sum of the squares of the individual velocities.

Answer 2

At 4:00 p.m., the distance between ship A and ship B is changing at a rate of 47.16 km/h, found by using the Pythagorean theorem and differentiating the distance function with respect to time.

Step-by-step explanation:

To calculate how fast the distance between the two ships is changing at 4:00 p.m., we'll use the concept of relative motion in two dimensions, applying the Pythagorean theorem to find the rate of change of distance between the two ships.

Let's define the positions of ship A and B based on the origin. At noon, ship A is at a position (-170 km, 0 km), since it is 170 km west (left on the x-axis) of ship B, which is at the origin (0 km, 0 km). Ship A is moving east at 40 km/h, and ship B is moving north at 25 km/h. After 4 hours (from noon to 4:00 p.m.), ship A will have moved 160 km east (4 hours × 40 km/h), and ship B will have moved 100 km north (4 hours × 25 km/h).

At 4:00 p.m., the position of ship A is (-170 km + 160 km, 0 km) = (-10 km, 0 km). The position of ship B is (0 km, 100 km). We can now use the Pythagorean theorem to find the distance between the two ships:

√((-10 km)^2 + (100 km)^2) = √(100 + 10000) = √(10100) = 100.5 km

However, since we want the rate at which this distance is changing, we need to differentiate the distance function with respect to time, considering the right-angled triangle formed by the paths of A and B. The rates of change of the sides of the triangle are 40 km/h (eastward for A) and 25 km/h (northward for B). By applying the chain rule, we can write the rate of change of the hypotenuse, represented as 'c', with respect to time 't'.

`(dc/dt)^2 = (dx/dt)^2 + (dy/dt)^2`

`(dc/dt)^2 = (40 km/h)^2 + (25 km/h)^2`

`dc/dt = √(1600 + 625) = √2225 km^2/h^2 = 47.16 km/h`

Therefore, at 4:00 p.m., the distance between the two ships is changing at a rate of 47.16 km/h.