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Answer: Hello there!
If we define the x-axis as the one parallel to the ground and y as the perpendicular ( or the height in this case) then you know that the y position of the helicopter is equal to 0.5 kilometers, and is constant.
And the x component is x = (2km/min)*t where t is in minutes, and we could define t = 0 when the helicopter is over the white house.
Also, in this case, we can define the position of the white house as y = 0 and x = 0
Then our position vector (x,y) for the helicopter is ( 2*t, 0.5) and the position of the white house is (0,0)
then the distance between the helicopter and the white house is:
[tex] d(t) =I(2*t,0.5) - (0,0)I = I(2*t, 0.5)I= \sqrt{ ((2*t)^2 +0.5^2)}[/tex]
where the distance is calculated as the magnitude of the difference in the position between the two objects
Now we want to know the rate of change in the distance when t = 3.
This is equivalent to calculate the derivative of d(t) valuated in t = 3 minutes, this is d'(3)
then we need to derivate our equation:
[tex]d'(t) = \frac{dd(t)}{dt} = \frac{d(\sqrt{ ((2*t)^2 +0.5^2)})}{dt} = 0.5*((2*t)^2 +0.5^2)^{-1/2} *4*t[/tex]
now we can replace t by 3, and see the value of the rate of change 3 minutes after the helicopter was directli over the white house:
[tex]d'(3) = 0.5*((2*3)^2 +0.5^2)^{-1/2} *4*3 = 0.99[/tex]
The rate of change of the distance is 0.99 kilometers per minute square.
Rate by which the distance between the helicopter and the White House changing is 0.99 km/m after 3 minutes of fly.
What is the rate of speed?
The rate of speed is the rate at which the total distance travelled in the time taken. Rate of speed can be given as,
[tex]s=\dfrac{d}{t}[/tex]
Here, (d) is the distance travelled by the object and (t) is time taken but the object to cover that distance.
The speed of the helicopter is 2 kilometers per minute. Change it into the meter per second as,
[tex]v=2\times\dfrac{1000}{60}\\v=33.33\rm m/s[/tex]
As, the speed of the helicopter is 33.33 m/s. Thus, the distance traveled by the helicopter in 1 minute is,
[tex]d=33.33\times3\\d=6000\rm m[/tex]
The height of the helicopter is 1/2 kilometer, which is equal to 500 meters. Thus, by the Pythagoras theorem the distance traveled by helicopter is,
[tex]h=\sqrt{6000^2+500^2}\\h=6020.8\rm m[/tex]
Hence, the rate by which the distance between the helicopter and the White House changing 3 minutes after the helicopter flies over the White House,
[tex]v=\dfrac{33.33\times3\times60}{6020.8}\\v=0.99\rm km/m[/tex]
Hence, the rate by which the distance between the helicopter and the White House changing is 0.99 km/m after 3 minutes of fly.
Learn more about the rate of speed here:
https://brainly.com/question/359790
