A rocket fired straight in the air is being tracked by a radar station 3 miles from the launching pad. If the rocket is travelling at 2 miles per second, how fast is the distance between the rocket and the tracking station changing at the moment when the rocket is 4 miles in the air?
D^2=9+y^2
I assume you meant fired straight up in the air. In that case, at time t seconds, the rocket has gone 2t miles.
So, the distance D is found, as you showed, by
D^2 = 9+y^2 = 9+4t^2
when the rocket is 4 miles up, t=2, and we have a 3-4-5 triangle, so D=5.
2D dD/dt = 8t
when t=2, we have
2(5) dD/dt = 8(2)
dD/dt = 8/5
To determine how fast the distance between the rocket and the tracking station is changing, we need to find the derivative of the equation that relates the distance and the height of the rocket.
The given equation is D^2 = 9 + y^2, where D is the distance between the rocket and the tracking station, and y is the height of the rocket.
To find the derivative, we can differentiate both sides of the equation with respect to time (t), using the chain rule:
d/dt (D^2) = d/dt (9 + y^2)
Applying the chain rule, we get:
2D * dD/dt = 0 + 2y * dy/dt
Simplifying the equation further:
2D * dD/dt = 2y * dy/dt
Now, we are given that the rocket is traveling at a rate of 2 miles per second, which means the rate of change of the height of the rocket (dy/dt) is 2.
Substituting the given value into the equation:
2D * dD/dt = 2y * 2
2D * dD/dt = 4y
Next, we need to find the value of D and y when the rocket is 4 miles in the air. From the equation D^2 = 9 + y^2, we can solve for D:
D^2 = 9 + y^2
D^2 = 9 + 4^2 (since y = 4 when the rocket is 4 miles in the air)
D^2 = 9 + 16
D^2 = 25
D = 5 (taking the positive square root)
Now that we have the values of D = 5 and y = 4, we can substitute them back into the equation:
2 * 5 * dD/dt = 4 * 4
10 * dD/dt = 16
Finally, we can solve for dD/dt (the rate at which the distance is changing) by dividing both sides of the equation by 10:
dD/dt = 16/10
Therefore, when the rocket is 4 miles in the air, the distance between the rocket and the tracking station is changing at a rate of 1.6 miles per second.
To find how fast the distance is changing, we need to find the derivative of the distance equation with respect to time.
Given: D^2 = 9 + y^2
Taking the derivative of both sides with respect to time (t), we get:
2D(dD/dt) = 0 + 2y(dy/dt)
Since the radar station is 3 miles away from the launching pad, we have D = 3 + y.
Substituting D = 3 + y into the derivative equation, we get:
2(3 + y)(dD/dt) = 2y(dy/dt)
Now, we can substitute the values given. At the moment when the rocket is 4 miles in the air, y = 4.
2(3 + 4)(dD/dt) = 2(4)(dy/dt)
Simplifying the equation, we have:
14(dD/dt) = 8(dy/dt)
Finally, divide both sides by 14 to solve for (dD/dt):
(dD/dt) = (8(dy/dt)) / 14
Simplifying further, we get:
(dD/dt) = 4(dy/dt) / 7
Therefore, the speed at which the distance between the rocket and the tracking station is changing is 4/7 times the speed at which the rocket is ascending.