At noon, ship A is 130 km west of ship B. Ship A is sailing east at 25 km/h and ship B is sailing north at 20 km/h. How fast is the distance between the ships changing at 4:00 PM?
At time t hours, we have the distance z between the ships as
z^2 = (130-25t)^2 + (20t)^2
at 4:00, z^2 = 30^2+80^2=7300, so z=10√73
Now,
2z dz/dt = 2(130-25t)(-25)+2(20t)(20)
z dz/dt = 1025t-3250
so, at t=4,
dz/dt = 85/√73 km/hr
Well, here's the situation: Ship A is sailing east, and ship B is sailing north. They are going in different directions, just like when you and your friend try to decide on a movie to watch! Ship A is going at 25 km/h and Ship B is going at 20 km/h.
Now let's crunch some numbers! At noon, the ships were 130 km apart. And from noon until 4:00 PM, 4 hours have passed. Since Ship A is sailing east, it has covered a distance of 25 km/h * 4 h = 100 km.
So, at 4:00 PM, Ship A is 130 km - 100 km = 30 km west of Ship B.
Now comes the fun part – calculating how fast the distance between the ships is changing. We can use the Pythagorean theorem (remember math?) to find out. The distance between the ships is like the hypotenuse of a right triangle.
The distance between the ships can be represented by D = sqrt(Dx^2 + Dy^2), where Dx is the distance traveled by Ship A in the x-direction (east) and Dy is the distance traveled by Ship B in the y-direction (north).
To find how fast the distance is changing, we can use the chain rule from calculus. We'll differentiate the equation D = sqrt(Dx^2 + Dy^2) with respect to time t.
dD/dt = (1/2) * (2Dx * dDx/dt + 2Dy * dDy/dt)
Since we know that dDx/dt = 25 km/h and dDy/dt = 20 km/h, plugging in these values, we get:
dD/dt = (1/2) * (2 * 100 * 25 + 2 * 0 * 20) = 2500 km/h
So, the distance between the ships is changing at a rate of 2500 km/h at 4:00 PM. That's pretty fast! I hope these calculations have sailed smoothly for you!
To find the rate at which the distance between the ships is changing at 4:00 PM, we can use the concept of related rates.
Let's define some variables:
- Distance between ships A and B at 4:00 PM: D(t)
- Time: t
We know that ship A is sailing east at a constant speed of 25 km/h, so the position of ship A at time t is given by:
x_A = 25t
Similarly, ship B is sailing north at a constant speed of 20 km/h, so the position of ship B at time t is given by:
y_B = 20t
Since the ships are moving in different directions, we can use the Pythagorean theorem to find the distance between them:
D(t) = √((x_A - x_B)^2 + (y_A - y_B)^2)
First, we need to find the values of x_B and y_A at 4:00 PM. Since ship A started 130 km west of ship B at noon, it has been traveling for 4 hours. Therefore, at 4:00 PM, ship A would have moved:
x_A = 25 * 4 = 100 km
Since ship B is sailing north at a constant speed of 20 km/h, it would have traveled:
y_B = 20 * 4 = 80 km
Now, we can substitute the values into the equation for D(t):
D(t) = √((100 - x_B)^2 + (y_A - 80)^2)
To find the rate at which D(t) is changing at 4:00 PM, we need to find dD/dt (the derivative of D with respect to t). Since dD/dt represents the rate of change of D with respect to t, it gives us the rate at which the distance between the ships is changing.
Differentiating D(t) with respect to t, we get:
dD/dt = (1/2) * (1/√((100 - x_B)^2 + (y_A - 80)^2)) * 2 * ((100 - x_B) * (-dx_B/dt)) + 2 * ((y_A - 80) * dy_A/dt)
Now, let's substitute the values we know:
x_B = 0 (since ship B was at the origin)
dx_B/dt = 0 (since ship B is not moving in the x-direction)
y_A = 20 * 4 = 80 km
dy_A/dt = 25 km/h (since ship A is moving at a constant speed of 25 km/h)
Substituting these values into the equation, we get:
dD/dt = (1/2) * (1/√((100 - 0)^2 + (80 - 80)^2)) * 2 * ((100 - 0) * 0) + 2 * ((80 - 80) * 25)
= 0 + 0
Therefore, the rate at which the distance between the ships is changing at 4:00 PM is 0 km/h.
To find the rate at which the distance between the two ships is changing, we need to use the concept of related rates.
Let's consider the situation at a specific time, say 4:00 PM. At that moment, ship A has been sailing east for 4 hours, and ship B has been sailing north for 4 hours as well.
First, let's find the positions of the two ships at 4:00 PM. Ship A has traveled 25 km/h for 4 hours, so it has moved 25 km/h * 4 h = 100 km to the east. Ship B has traveled 20 km/h for 4 hours, so it has moved 20 km/h * 4 h = 80 km to the north.
Now we can draw a diagram to illustrate the positions of the two ships at 4:00 PM.
A(100 km, 0 km)
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B(0 km, 80 km)
Now, we can calculate the distance between the ships using the distance formula (Pythagorean theorem) at 4:00 PM. The distance between the two ships is given by the formula:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
In this case, (x1, y1) represents the position of Ship A (100 km, 0 km) and (x2, y2) represents the position of Ship B (0 km, 80 km). Plugging in these values into the formula, we get:
distance = √((0 km - 100 km)^2 + (80 km - 0 km)^2)
= √((-100 km)^2 + (80 km)^2)
= √(10000 km^2 + 6400 km^2)
= √(16400 km^2)
≈ 128.06 km
Now, to find how fast the distance between the ships is changing at 4:00 PM, we need to calculate the derivative of the distance with respect to time. Let's call this rate of change "d(distance)/dt".
Using the chain rule, we can express d(distance)/dt in terms of the rates of change of x and y coordinates with respect to time.
d(distance)/dt = (d(distance)/dx) * (dx/dt) + (d(distance)/dy) * (dy/dt)
Since the rate of change of y (dy/dt) is given as 20 km/h (northward) and the rate of change of x (dx/dt) is given as 25 km/h (eastward), we can calculate:
d(distance)/dt = (d(distance)/dx) * (25 km/h) + (d(distance)/dy) * (20 km/h)
To find d(distance)/dx and d(distance)/dy, we differentiate the distance formula with respect to x and y, respectively.
d(distance)/dx = (2*(x2 - x1))/2 sqrt((x2 - x1)^2 + (y2 - y1)^2)
= (x2 - x1)/(distance)
d(distance)/dy = (2*(y2 - y1))/2 sqrt((x2 - x1)^2 + (y2 - y1)^2)
= (y2 - y1)/(distance)
Plugging in the values we have calculated, we get:
d(distance)/dt = ((100 km)/(128.06 km)) * (25 km/h) + ((80 km)/(128.06 km)) * (20 km/h)
= 19.53 km/h + 15.73 km/h
≈ 35.26 km/h
Therefore, at 4:00 PM, the distance between the ships is changing at a rate of approximately 35.26 km/h.