Question:

A hot air balloon is rising straight up from the ground. At a certain moment, the distance between the balloon and an observer on the ground is 500 feet, and it is increasing at a rate of 200 feet per minute. At the same time, the angle of elevation from the observer to the balloon is increasing at a rate of 0.03 radians per minute. How fast is the balloon rising?

Assume the triangle formed by the balloon, observer, and ground is a right triangle.

Solution:

Let's define the variables:

• Let x be the distance between the observer and the balloon (in feet).
• Let θ be the angle of elevation from the observer to the balloon (in radians).
• Let h be the height of the balloon above the ground (in feet).

We are given that dx/dt = 200 ft/min (rate of change of distance between balloon and observer) and dθ/dt = 0.03 rad/min (rate of change of angle of elevation).

We need to find dh/dt (rate of change of height of the balloon).

By trigonometry, we can express h in terms of x and θ:

h = x * tan(θ)

Now, we differentiate both sides of this equation with respect to time t:

(dh/dt) = (dx/dt) * tan(θ) + x * sec^2(θ) * (dθ/dt)

Substituting the given values, we have:

(dh/dt) = (200 ft/min) * tan(θ) + x * sec^2(θ) * (0.03 rad/min)

We need to find dh/dt when x = 500 ft and θ is the angle that satisfies the right triangle formed by the balloon, observer, and ground.

Since x = 500 ft, we need to find the value of θ. To do this, let's consider the triangle formed:

   /|\
  / | \
 /  |θ\
/___|___\
 500 ft

Since we have a right triangle, we can use the inverse tangent function:

θ = arctan(h/500)

Plugging in the values for h = 500 ft, we find:

θ = arctan(500/500) = arctan(1) = π/4 radians (45°)

Now we can substitute these values into the equation for (dh/dt):

(dh/dt) = (200 ft/min) * tan(π/4) + (500 ft) * sec^2(π/4) * (0.03 rad/min)

Using the trigonometric identities tan(π/4) = 1 and sec^2(π/4) = 2, we simplify the equation:

(dh/dt) = (200 ft/min) + (500 ft) * 2 * (0.03 rad/min)

(dh/dt) = 200 ft/min + (500 ft) * 0.06 rad/min

(dh/dt) = 200 ft/min + 30 ft/min

(dh/dt) = 230 ft/min

Therefore, the balloon is rising at a rate of 230 ft/min.