Question

A red ballon is 40 feet above the ground and rising at 2 ft/s. At the same time, a blue balloon is at 60 feet above the ground and descending at 3 ft/s. What will the height of the balloons be when they are the same height above the ground

Answer 1

Answer: 48 ft

Explanation:

The height gap between the balloons is 60 -40 = 20 feet. That gap is being closed at the rate of 2 + 3 = 5 ft/s, so will be gone in ...

(20 ft)/(5 ft/s) = 4 s

At that time, the red balloon will have risen (2 ft/s)(4 s) = 8 ft to a height of ...

40 ft +8 ft = 48 ft

The blue balloon will have descended (3 ft/s)(4 s) = 12 ft to a height of ...

60 ft -12 ft = 48 ft

The balloons at at 48 ft when they are both the same height.

_____

Time and speed and distance are related by the formula you see on every speed limit sign:

speed = distance/time . . . . . . . (on the sign, it's "miles per hour")

or

time = distance/speed

or

distance = speed × time

_____

If you want equations, you can write them as ...

h = 40 +2t

h = 60 -3t

where h is the altitude the balloons have when they are at the same height, and t is the number of seconds it takes to get there.

We're only interested in h, so we can cancel t by multiplying the first equation by 3 and adding that to the second equation multiplied by 2:

3(h) + 2(h) = 3(40 +2t) +2(60 -3t)

5h = 120 +6t +120 -6t

h = 240/5 = 48 . . . . the height in feet at which the balloons are the same height