Two points, A and B, are 275 ft apart. At a given instant, a balloon is released
at B and rises vertically at a constant rate of 2.5 ft/sec, and, at the same
instant, a cat starts running from A to B at a constant rate of 5 ft/sec.
a) After 40 seconds, is the distance between the cat and the balloon
decreasing or increasing? At what rate?
b) Describe what is happening to the distance and the balloon at 50 t =
seconds.
To answer these questions, we need to analyze the motion of both the cat and the balloon.
a) After 40 seconds, let's find the position of the cat and the balloon:
Distance covered by the cat = Speed of the cat * Time = 5 ft/sec * 40 sec = 200 ft.
Distance covered by the balloon = Speed of the balloon * Time = 2.5 ft/sec * 40 sec = 100 ft.
Now let's calculate the distance between the cat and the balloon:
Distance = Distance between A and B - Distance covered by the balloon
= 275 ft - 100 ft
= 175 ft.
Since the distance between the cat and the balloon after 40 seconds is 175 ft, and the initial distance between them was 275 ft, we can conclude that the distance is decreasing.
To determine the rate at which the distance is decreasing, we can calculate the relative velocity of the cat with respect to the balloon:
Relative velocity = Speed of the cat - Speed of the balloon
= 5 ft/sec - 2.5 ft/sec
= 2.5 ft/sec.
Therefore, the distance between the cat and the balloon is decreasing at a rate of 2.5 ft/sec.
b) To describe what is happening at t = 50 seconds, we need to find the positions of the cat and the balloon:
Distance covered by the cat = Speed of the cat * Time = 5 ft/sec * 50 sec = 250 ft.
Distance covered by the balloon = Speed of the balloon * Time = 2.5 ft/sec * 50 sec = 125 ft.
Now let's calculate the distance between the cat and the balloon:
Distance = Distance between A and B - Distance covered by the balloon
= 275 ft - 125 ft
= 150 ft.
From the calculations, we can see that at t = 50 seconds, the distance between the cat and the balloon is 150 ft. Therefore, the distance has decreased compared to the initial distance of 275 ft.
It's important to note that at t = 50 seconds, the balloon has risen higher vertically, but its horizontal position remains the same because it was released at point B. So, the balloon is directly above point B, 150 ft above the ground.