Answer:
0.0830 radians per second
Step-by-step explanation:
If θ is the angle of elevation and h is the height of the balloon, then:
tan θ = h / 53
Take the derivative of both sides with respect to time:
sec² θ dθ/dt = 1/53 dh/dt
Use Pythagorean identity:
(1 + tan² θ) dθ/dt = 1/53 dh/dt
Substitute:
(1 + (h/53)²) dθ/dt = 1/53 dh/dt
Multiply both sides by 53²:
(53² + h²) dθ/dt = 53 dh/dt
Given dh/dt = 6 and h = 32:
(53² + 32²) dθ/dt = 53 (6)
dθ/dt = 318 / 3833
dθ/dt ≈ 0.0830
The angle of elevation is increasing at a rate of 0.0830 radians per second.
