Final answer:
It will take approximately 142 seconds for Keith's hot air balloon to reach the same height above the ground as Erica's hot air balloon.
Step-by-step explanation:
To determine how many seconds it will take for Keith's hot air balloon to be the same number of feet above the ground as Erica's, we should set their distances above ground equal to each other and solve for the time t.
Let's say Keith's height above the ground at any time t is represented by the equation: Hk(t) = 200 + 1.2t, since he is ascending at 1.2 ft./s from an initial height of 200 feet. Erica's height above the ground at any time t can be represented as He(t) = 548 - 1.25t, because she is descending at 1.25 ft./s from an initial height of 548 feet.
To find out when Keith will be at the same height as Erica, we can set Hk(t) equal to He(t):
200 + 1.2t = 548 - 1.25t
Combining like terms, we get:
1.2t + 1.25t = 348
2.45t = 348
t = 348 / 2.45
t ≈ 142 seconds
So, it will take Keith approximately 142 seconds for his balloon to be the same number of feet above the ground as Erica's balloon.