an observer is standing 200 meters from a hot air ballon, when is rising vertically at constant rate of 4m/sec. how fast is the angle of elevation changing 30 seconds after launch?
To find the rate at which the angle of elevation is changing, we can use trigonometry. Let A be the height of the hot air balloon and θ be the angle of elevation.
We have the following information:
- The distance between the observer and the hot air balloon is given as 200 meters.
- The hot air balloon is rising vertically at a constant rate of 4 m/sec.
We want to find how fast the angle of elevation is changing 30 seconds after launch.
Let's start by setting up a trigonometric relationship involving the angle of elevation. In a right triangle, the tangent of an angle is equal to the ratio of the opposite side (A) to the adjacent side (200 meters).
Given the tangent relationship, we have:
tan(θ) = A / 200
Differentiating both sides of the equation with respect to time (t), we get:
sec^2(θ) * dθ/dt = dA/dt / 200
We are given that dA/dt = 4 m/sec (the rate at which the hot air balloon is rising).
Substituting the values, we have:
sec^2(θ) * dθ/dt = 4 / 200
Now, we need to find the value of sec^2(θ) (the reciprocal of the cosine squared of θ).
Since we are given the distance (200 meters), we can use the Pythagorean theorem to find the height of the balloon at any given time. Let h be the height of the balloon at any time t.
According to the Pythagorean theorem, we have:
h^2 + 200^2 = A^2
Differentiating both sides of the equation with respect to time (t), we get:
2h * dh/dt = 2A * dA/dt
Rearranging the equation, we have:
dh/dt = (A/A) * (dA/dt) / h
Since tan(θ) = A / 200, we can rewrite the equation as:
dh/dt = (200 tan(θ) * dA/dt) / h
Substituting the given values, we have:
dh/dt = (200 * tan(θ) * 4) / h
dh/dt = (800 tan(θ)) / h
Now, we can substitute this expression for dh/dt in the previous equation:
sec^2(θ) * dθ/dt = 4 / 200
sec^2(θ) * dθ/dt = 1 / 50
From this equation, we can solve for dθ/dt by multiplying both sides by 50 and taking the square root:
dθ/dt = sqrt(1 / (50 sec^2(θ)))
To find the value of sec(θ), we can use the relationship between the height of the balloon and the distance between the observer and the balloon:
sec(θ) = A / 200
Given that the observer is standing 200 meters from the balloon, we can substitute A = h + 200:
sec(θ) = (h + 200) / 200
Substituting this expression for sec(θ) into the equation for dθ/dt, we have:
dθ/dt = sqrt(1 / (50 * ((h + 200) / 200)^2))
Now, we can substitute the given values into this expression to find dθ/dt 30 seconds after launch.
To find how fast the angle of elevation is changing, we need to use basic trigonometry. Let's break down the problem step by step.
Step 1: Understand the problem
An observer is standing 200 meters away from a hot air balloon that is rising vertically at a constant rate of 4 m/sec. We want to determine how fast the angle of elevation is changing 30 seconds after the balloon is launched.
Step 2: Define the relevant variables
Let's define the following variables:
- The distance between the observer and the hot air balloon: d = 200 meters
- The vertical rate at which the balloon is rising: h' = 4 m/sec
- The angle of elevation, which we need to find: θ (theta)
- The time after the balloon is launched: t = 30 seconds
Step 3: Establish a relation between the variables
Using trigonometry, we know that the tangent of the angle of elevation is equal to the ratio of the vertical distance (height of the balloon above the observer) to the horizontal distance (distance between the observer and the balloon). In this case, the tangent of θ is given by:
tan(θ) = (height of the balloon) / (distance between the observer and the balloon)
Step 4: Find the vertical distance
The vertical distance or the height of the balloon above the observer can be determined using the given vertical rate:
height of the balloon = (vertical rate) × (time after launch)
height of the balloon = h' × t
Plugging in the given values, we get:
height of the balloon = 4 m/sec × 30 sec
height of the balloon = 120 meters
Step 5: Calculate the angle of elevation
Now we can substitute the values we have into the tangent equation to find the angle of elevation:
tan(θ) = (height of the balloon) / (distance between the observer and the balloon)
tan(θ) = 120 meters / 200 meters
Using a calculator, we find that:
tan(θ) = 0.6
Step 6: Find the rate of change of the angle of elevation
To find how fast the angle of elevation is changing, we need to calculate the derivative of the tangent function, with respect to time:
d(tan(θ)) / dt = d/dt (height of the balloon) / (distance between the observer and the balloon)
To simplify, we can use the quotient rule:
d(tan(θ)) / dt = [(d(height of the balloon)/dt)*(distance between the observer and the balloon) - (height of the balloon)*(d(distance between the observer and the balloon)/dt)] / (distance between the observer and the balloon)^2
Since the balloon is rising vertically and the observer is stationary, the distance between the observer and the balloon remains constant (200 meters). Therefore, the derivative of the distance with respect to time is zero:
d(distance between the observer and the balloon)/dt = 0
Simplifying the equation further, we have:
d(tan(θ)) / dt = (d(height of the balloon)/dt) / (distance between the observer and the balloon)
Plugging in the relevant values, we get:
d(tan(θ)) / dt = (4 m/sec) / (200 meters)
Calculating this value, we find:
d(tan(θ)) / dt = 0.02 rad/sec
Therefore, the angle of elevation is changing at a rate of 0.02 radians per second, 30 seconds after the balloon's launch.