Answer :
To determine the correct system of equations representing the heights of two hot air balloons after [tex]\( m \)[/tex] minutes, we need to carefully interpret the information given:
1. One balloon started at a height of 3000 feet and is decreasing in height at a rate of 40 feet per minute.
2. The second balloon is rising at a rate of 50 feet per minute after beginning from a height of 1200 feet.
Considering this information, we can express the height [tex]\( h \)[/tex] of each balloon after [tex]\( m \)[/tex] minutes using linear equations:
### Balloon 1:
- Initial height: 3000 feet
- Decreasing at a rate of 40 feet per minute
The equation for Balloon 1's height [tex]\( h \)[/tex] after [tex]\( m \)[/tex] minutes is:
[tex]\[ h = 3000 - 40m \][/tex]
### Balloon 2:
- Initial height: 1200 feet
- Rising at a rate of 50 feet per minute
The equation for Balloon 2's height [tex]\( h \)[/tex] after [tex]\( m \)[/tex] minutes is:
[tex]\[ h = 1200 + 50m \][/tex]
Now let's look at the given options and compare them with our derived equations:
A.
[tex]\[
\begin{array}{l}
h = 3000m - 40 \\
h = 1200m + 60
\end{array}
\][/tex]
This option is incorrect because it does not match our derived equations at all and includes incorrect coefficients and operations.
B.
[tex]\[
\begin{array}{l}
h = 3000 + 40m \\
h = 1200 + 50m
\end{array}
\][/tex]
This option is also incorrect because it suggests that the height of Balloon 1 is increasing rather than decreasing.
C.
[tex]\[
\begin{array}{l}
h = 3000 = 40m \\
h = 1200 + 50m
\end{array}
\][/tex]
This option is incorrect because it represents Balloon 1's height equation incorrectly with an equal sign rather than a subtraction operation.
D.
[tex]\[\begin{array}{r}
m = 3000 - 40n \\
m = 1200 + 50h
\end{array}\][/tex]
This option is incorrect because it misrepresents the variables in the equations.
Given the detaile reviewed, none of the given options are correct. However, considering the feedback, it seems that the correct system of equations should be:
[tex]\[
\begin{array}{l}
h = 3000 - 40m \\
h = 1200 + 50m
\end{array}
\][/tex]
1. One balloon started at a height of 3000 feet and is decreasing in height at a rate of 40 feet per minute.
2. The second balloon is rising at a rate of 50 feet per minute after beginning from a height of 1200 feet.
Considering this information, we can express the height [tex]\( h \)[/tex] of each balloon after [tex]\( m \)[/tex] minutes using linear equations:
### Balloon 1:
- Initial height: 3000 feet
- Decreasing at a rate of 40 feet per minute
The equation for Balloon 1's height [tex]\( h \)[/tex] after [tex]\( m \)[/tex] minutes is:
[tex]\[ h = 3000 - 40m \][/tex]
### Balloon 2:
- Initial height: 1200 feet
- Rising at a rate of 50 feet per minute
The equation for Balloon 2's height [tex]\( h \)[/tex] after [tex]\( m \)[/tex] minutes is:
[tex]\[ h = 1200 + 50m \][/tex]
Now let's look at the given options and compare them with our derived equations:
A.
[tex]\[
\begin{array}{l}
h = 3000m - 40 \\
h = 1200m + 60
\end{array}
\][/tex]
This option is incorrect because it does not match our derived equations at all and includes incorrect coefficients and operations.
B.
[tex]\[
\begin{array}{l}
h = 3000 + 40m \\
h = 1200 + 50m
\end{array}
\][/tex]
This option is also incorrect because it suggests that the height of Balloon 1 is increasing rather than decreasing.
C.
[tex]\[
\begin{array}{l}
h = 3000 = 40m \\
h = 1200 + 50m
\end{array}
\][/tex]
This option is incorrect because it represents Balloon 1's height equation incorrectly with an equal sign rather than a subtraction operation.
D.
[tex]\[\begin{array}{r}
m = 3000 - 40n \\
m = 1200 + 50h
\end{array}\][/tex]
This option is incorrect because it misrepresents the variables in the equations.
Given the detaile reviewed, none of the given options are correct. However, considering the feedback, it seems that the correct system of equations should be:
[tex]\[
\begin{array}{l}
h = 3000 - 40m \\
h = 1200 + 50m
\end{array}
\][/tex]
