a hot-air balloon rising srtaight up from level field is tracked by a range finder 5000 feet from the liftoff point. At the moment the range finder's elevation angle is 45 degrees (pie/4 radians), the angle is increasing at the rate of 0.14 radians per minute. how fast is the balloon rising at that moment?
altitude=5000*sinTheta
d altitude/dt= 5000*cosTheta* dTheta/dt
To find the rate at which the balloon is rising, we need to differentiate the equation that relates the distance between the range finder and the balloon with respect to time. Let's call the height of the balloon "h" and the distance between the range finder and the balloon "d".
Using trigonometry, we know that tan(theta) = h/d, where theta is the elevation angle. We need to find dh/dt, the rate at which the height is changing when the range finder's elevation angle is 45 degrees.
First, differentiate both sides of the equation with respect to time (t):
d(tan(theta))/dt = d(h/d)/dt
Using the chain rule, we get:
sec^2(theta) * dtheta/dt = (1/d) * (dh/dt)
Since the range finder is 5000 feet from the liftoff point, the distance can be expressed as d = sqrt(h^2 + 5000^2).
Now, plugging in the given values:
- theta = π/4 radians
- dtheta/dt = 0.14 radians per minute
We can find d(theta)/dt by finding sec^2(π/4) = 2.
Substituting the values into the equation, we get:
2 * 0.14 = (1/sqrt(h^2 + 5000^2)) * (dh/dt)
Simplifying:
0.28 = (1/sqrt(h^2 + 5000^2)) * (dh/dt)
To find dh/dt, we need to isolate it:
dh/dt = 0.28 * sqrt(h^2 + 5000^2)
This equation gives us the rate at which the balloon is rising at the moment when the range finder's elevation angle is 45 degrees (π/4 radians). We can plug in any value for h to find the rate at different heights.