At noon of a certain day,ship A is 60 km due north of ship B. If A sails east at 12 km/hr and B sails north 9 km/hr,determine how rapidly the distance between them is changing 2 hrs later. Is it increasing or decreasing?
2d dd/dt = 450t-1080
dd/dt = 900-1080/(2*48.37)=-1.86
this is what I get when I substitute the values.Is it correct?
To determine how rapidly the distance between the two ships is changing, we can use the concept of rate of change and the Pythagorean theorem.
Let's denote the distance between the two ships as D(t), where t represents time.
After 2 hours, ship A will have traveled a distance of 12 km/hr * 2 hrs = 24 km to the east. Thus, the position of ship A is now 60 km + 24 km = 84 km due north of ship B.
Using the Pythagorean theorem, we can calculate the distance between the two ships at this new position:
D(t) = √((distance north)^2 + (distance east)^2)
D(t) = √((84 km)^2 + (24 km)^2)
D(t) = √(7056 km^2 + 576 km^2)
D(t) = √(7632 km^2)
D(t) ≈ 87.36 km
Now, to find how rapidly the distance between the two ships is changing, we can calculate the derivative of D(t) with respect to time.
dD/dt = (d/dt) √(7056 km^2 + 576 km^2)
The derivative of √(7056 km^2 + 576 km^2) can be found using the chain rule:
dD/dt = (1/2) * (7056 km^2 + 576 km^2)^(-1/2) * (2 * 7056 km^2 * (d/dt) + 2 * 576 km^2 * (d/dt))
Simplifying the expression, we get:
dD/dt = (1/2) * (7056 km^2 + 576 km^2)^(-1/2) * (2 * 7056 km^2 * 0 + 2 * 576 km^2 * 9 km/hr)
dD/dt = (1/2) * (7056 + 576)^(-1/2) * (2 * 576 * 9)
dD/dt = (1/2) * (7632)^(-1/2) * (10368)
dD/dt ≈ 0.1 km/hr * 10368
dD/dt ≈ 1036.8 km/hr
Therefore, the distance between the two ships is changing at a rate of approximately 1036.8 km/hr. Since the rate is positive, the distance between the two ships is increasing.
To determine how rapidly the distance between the two ships is changing, we'll use the derivative.
Let's start by drawing a diagram to visualize the situation. We have ship A and ship B, with A initially 60 km due north of B.
A
|
|
60 km |
--------- B
Let's assume that at time t = 0, ship A is at the point (0, 60) and ship B is at the point (0, 0).
Since ship A is sailing east at 12 km/hr, after 2 hours, it will have traveled a horizontal distance of 12 km/hr * 2 hr = 24 km. Therefore, after 2 hours, ship A will be at the point (24, 60).
Since ship B is sailing north at 9 km/hr, after 2 hours, it will have traveled a vertical distance of 9 km/hr * 2 hr = 18 km. Therefore, after 2 hours, ship B will be at the point (0, 18).
B
|
|
18 km |
--------- A
|
|
|
|
| 24 km
Now, let's calculate the distance between ship A and ship B after 2 hours. We can use the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Here, (x1, y1) = (0, 18) and (x2, y2) = (24, 60).
distance = sqrt((24 - 0)^2 + (60 - 18)^2)
= sqrt(24^2 + 42^2)
= sqrt(576 + 1764)
= sqrt(2340)
= 48.39 km (approximately)
Now, let's find out how rapidly the distance between the ships is changing. We want to find d(distance)/dt, the rate of change of the distance with respect to time.
To do this, we'll consider ship A and ship B as moving points. We know that ship A is moving horizontally at a rate of 12 km/hr, and ship B is moving vertically at a rate of 9 km/hr.
Let x be the horizontal distance between the ships, and y be the vertical distance between them. The distance between the ships can be calculated as:
distance = sqrt(x^2 + y^2)
To find the rate of change with respect to time, we'll differentiate the equation with respect to t:
d(distance)/dt = d(sqrt(x^2 + y^2))/dt
Using the chain rule, the equation becomes:
d(distance)/dt = (d(sqrt(x^2 + y^2))/dx)*(dx/dt) + (d(sqrt(x^2 + y^2))/dy)*(dy/dt)
Using the fact that dx/dt = 12 km/hr and dy/dt = 9 km/hr, we get:
d(distance)/dt = (x/sqrt(x^2 + y^2)) * 12 + (y/sqrt(x^2 + y^2)) * 9
Now we substitute the values we have at t = 2 hours:
x = 24 km (horizontal distance)
y = 42 km (vertical distance)
d(distance)/dt = (24/sqrt(24^2 + 42^2)) * 12 + (42/sqrt(24^2 + 42^2)) * 9
Simplifying this expression, we get:
d(distance)/dt = (24/48.39) * 12 + (42/48.39) * 9
d(distance)/dt = 12.47 km/hr
Therefore, the distance between the two ships is changing at a rate of approximately 12.47 km/hr after 2 hours.
Since the distance is increasing (because the two ships are moving away from each other), the rate is positive.
If we call A's position at noon (0,0) then we have the positions of A and B after t hours
A: (12t,0)
B: (0,60-9t)
The distance is thus
d^2 = (60-9t)^2 + (12t)^2 = 225t^2-1080t+3600
At t=2, d=√(42^2+24^2)=√2340=48.37
So, at any time t,
2d dd/dt = 450t-1080
Now just plug in your values