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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. Given that [tex]$h$[/tex] is the height of the balloons after [tex]$m$[/tex] minutes, determine which system of equations represents this situation.
A. [tex]\[
\begin{array}{l}
h = 3,000 - 40m \\
h = 1,200 + 50m
\end{array}
\][/tex]
B. [tex]\[
\begin{array}{l}
m = 3,000 - 40h \\
m = 1,200 + 50h
\end{array}
\][/tex]
C. [tex]\[
h = 3,000m - 40
\][/tex] [tex]\[
h = 1,200m + 50
\][/tex]
D. [tex]\[
\begin{array}{l}
h = 3,000 + 40m \\
h = 1,200 - 50m
\end{array}
\][/tex]
To solve the problem of representing the heights of the two hot air balloons after a certain number of minutes using a system of equations, let's analyze the given conditions step-by-step:
1. First Balloon: - Initial height: 3,000 feet. - Descending at a rate of 40 feet per minute.
Therefore, the height [tex]\( h_1 \)[/tex] of the first balloon after [tex]\( m \)[/tex] minutes can be represented by the equation: [tex]\[
h_1 = 3000 - 40m
\][/tex]
2. Second Balloon: - Initial height: 1,200 feet. - Rising at a rate of 50 feet per minute.
Therefore, the height [tex]\( h_2 \)[/tex] of the second balloon after [tex]\( m \)[/tex] minutes can be represented by the equation: [tex]\[
h_2 = 1200 + 50m
\][/tex]
Based on these representations, we need to identify which system of equations (from the provided options) correctly matches the equations we derived:
- Option A: [tex]\[
\begin{array}{l}
h = 3000 - 40m \\
h = 1200 + 50m
\end{array}
\][/tex] This matches our derived equations perfectly.
- Option B: [tex]\[
\begin{array}{l}
m = 3000 - 40h \\
m = 1200 + 50h
\end{array}
\][/tex] These equations incorrectly use [tex]\( h \)[/tex] and [tex]\( m \)[/tex] interchanged and have the wrong functional forms.
- Option C: [tex]\[
h = 3000m - 40 \\
h = 1200m + 50
\][/tex] These equations incorrectly place [tex]\( m \)[/tex] in a position to be multiplied by the initial heights, which is incorrect.
- Option D: [tex]\[
\begin{array}{l}
h = 3000 + 40m \\
h = 1200 - 50m
\end{array}
\][/tex] These equations indicate the wrong directions for the changes in height.
After evaluating all the options, we see that the correct system of equations is given in Option A. Therefore, the correct choice is:
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