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An observer stands 2400 ft away from a launch pad to observe a rocket launch. The rocket blasts off and maintains a velocity of 700 ft/sec. Assume the scenario can be modeled as a right triangle. How fast is the angle of elevation (in radians/sec) from the observer to rocket changing when the rocket is 700 ft from the ground

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Final answer:

The rate at which the angle of elevation changes as a rocket ascends can be determined using trigonometry and calculus' related rates. The observer is standing 2400 ft away from the launch pad, and when the rocket is 700 ft high, the derivative of the tangent function, along with the constant velocity of the rocket, is used to solve for the rate of change of the angle in radians per second.

Step-by-step explanation:

The question involves calculating how fast the angle of elevation is changing when a rocket is 700 ft from the ground. This scenario is described by a right triangle, with the rocket's altitude as one leg and the observer's distance from the launch pad as the other leg. To find the rate at which the angle of elevation is changing, we use the concepts of trigonometry and related rates from calculus.

Let's denote the height of the rocket as h, the horizontal distance of the observer from the launch pad as x (which is 2400 ft and remains constant), and the angle between the line of sight from the observer to the rocket and the ground as θ. The relationship between the height of the rocket and the angle of elevation is given by the tangent function: tan(θ) = h/x. When the rocket is 700 ft high, tan(θ) = 700/2400. We need to find dθ/dt, the rate of change of the angle of elevation with respect to time.

Using the derivative of the tangent function with respect to time, we get:

(d/dt)[tan(θ)] = (d/dt)[h/x]
sec2(θ) * (dθ/dt) = (dh/dt) / x

Given that the rocket's velocity (dh/dt) is 700 ft/sec, and x is 2400 ft, we can plug these values into the equation to solve for (dθ/dt).

sec2(θ) * (dθ/dt) = 700 / 2400

First, calculate sec(θ) when h = 700 ft. tan(θ) = 700/2400, θ = arctan(700/2400).

Once we have θ, we can find sec(θ) = 1/cos(θ). Then plug sec(θ) into the previous equation to solve for dθ/dt.

User Ayoub Laazazi
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4 votes

Answer:

d∅/dt = 0.2688radians/sec

Step-by-step explanation:

See attachment.

An observer stands 2400 ft away from a launch pad to observe a rocket launch. The-example-1
User Youcef
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