A hot air balloon rising vertically is tracked by an observer located 4 km from the lift‑off point. At a certain moment, the angle between the observer's line of sight and the horizontal is π/5, and it is changing at a rate of 0.4 rad/min. How fast is the balloon rising at this moment? Let y be the height of the balloon (in km), t be time (in minutes), and θ the angle between the line‑of‑sight and the horizontal (in radians).

1.22 km/min

Step-by-step explanation:

Let Q be baloon height at a time t. Our goal is to determine the speed of the baloon at the moment.

The dy / dt velocity of the baloon when = π/5?  

So we can restate the question as follows:

Owing to the fact d / dt = 0.2 rad / min at some stage = π/5

from fig:

tanθ = y/4

Differentiating w.r.t "t"

sec2 θ * dθ/dt = 1/4(dy/dt)

=> dy/dt = (4/cos2 θ)dθ/dt

At the given moment θ =  and dθ/dt = 0.2 rad/min.

dy/dt = (4/cos2)* (0.2)

= 1.22 km/min

And the velocity of the baloon currently is 1.22 km / min.

1) c 2143.6 cm

2) c 113.0 m

3) a 33.5 ft

4)   d 2 cm

5) c 20 mm

6) 180 in

7) d 120 ft

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for