Two hot air balloons are flying above a park. One balloon started at a height of 3,000 feet above the ground and is decreasing in height at a rate of 40 feet per minute. The second balloon is rising at a rate of 50 feet per minute after beginning from a height of 1,200 feet above the ground.

Question

Given that h is the height of the balloons after m minutes, determine which system of equations represents this situation.

A.
m = 3,000 - 40h
m = 1,200 + 50h
B.
h = 3,000 - 40m
h = 1,200 + 50m
C.
h = 3,000 + 40m
h = 1,200 - 50m
D.
h = 3,000m - 40
h = 1,200m + 50

Answer 1

To model the situation of the two hot air balloons, we define \( h \) as the height of each balloon after \( m \) minutes.

1. For the first balloon:

   • It starts at a height of 3,000 feet.
   • It is decreasing in height at a rate of 40 feet per minute.
   • Therefore, the height of this balloon after \( m \) minutes can be expressed as: \[ h = 3000 - 40m \]

2. For the second balloon:

   • It starts at a height of 1,200 feet.
   • It is rising at a rate of 50 feet per minute.
   • Therefore, the height of this balloon after \( m \) minutes can be expressed as: \[ h = 1200 + 50m \]

Putting these two equations together, we find that the system of equations representing the situation is: \[ h = 3000 - 40m \] \[ h = 1200 + 50m \]

Now we can compare this system with the given options:

• Option A: \( m = 3,000 - 40h \), \( m = 1,200 + 50h \)
• Option B: \( h = 3,000 - 40m \), \( h = 1,200 + 50m \) (matches our equations)
• Option C: \( h = 3,000 + 40m \), \( h = 1,200 - 50m \)
• Option D: \( h = 3,000m - 40 \), \( h = 1,200m + 50 \)

Thus, the correct answer is Option B: \[ h = 3,000 - 40m \] \[ h = 1,200 + 50m \]