A rocker arises from its launching pad with a velocity given. By 0.1 h m/s ,where h m is the height reached by the rocket at that time.the rocket is obserrvved from 2000 m away from the launch pad.at. What rate is the distance of the roocket from observer channging when the rocket is at an altitude of 5000 m?
The answer is suppose to be 464 m/s and please provide the whole procedure .Thanks.
To solve this problem, we can use related rates. We are given the height reached by the rocket at a certain time and the distance of the observer from the launch pad. We need to find the rate at which the distance of the rocket from the observer is changing when the rocket is at an altitude of 5000 m.
Let's set up our variables:
h = height reached by the rocket at a certain time (in meters)
v = velocity of the rocket (in m/s)
d = distance of the rocket from the observer (in meters)
We are given that the velocity of the rocket is changing at a rate of 0.1 h m/s. This means dv/dt = 0.1h.
We need to find dd/dt, the rate at which the distance of the rocket from the observer is changing. Since we know the distance between the observer and the launch pad is constant at 2000 m, we have a right triangle formed by the observer, the rocket, and the launch pad. We can use the Pythagorean theorem to relate h, d, and 2000.
Applying the Pythagorean theorem, we have:
h^2 + d^2 = 2000^2
Differentiate both sides with respect to time (t):
2h dh/dt + 2d dd/dt = 0
Since we are interested in finding dd/dt when h = 5000 m, we can substitute the given information into the equation.
When h = 5000, we know that h = 5000 and d = 2000, so we have:
(2)(5000)(0.1h) + (2)(2000)(dd/dt) = 0
Simplifying the equation:
10000h + 4000(dd/dt) = 0
dd/dt = -10000h/4000
dd/dt = -5h
Now, we can substitute h = 5000 into the equation:
dd/dt = -5(5000)
dd/dt = -25000 m/s
Since we are interested in the rate at which the distance of the rocket from the observer is changing, the negative sign indicates that the distance is decreasing. To get the magnitude, we take the absolute value:
|dd/dt| = |-25000| = 25000 m/s
Therefore, the rate at which the distance of the rocket from the observer is changing when the rocket is at an altitude of 5000 m is 25000 m/s.