Answer :
To determine the correct situation that matches the given system of equations, let's analyze them step-by-step.
The system of equations:
[tex]\[ \left\{\begin{array}{l} 13(v-w)=52 \\ 12(v+w)=72 \end{array}\right. \][/tex]
Here, [tex]\( v \)[/tex] represents the speed of the airplane in still air, and [tex]\( w \)[/tex] represents the speed of the wind.
1. First Equation Interpretation:
[tex]\[ 13(v-w)=52 \][/tex]
This equation signifies that Andy flew the airplane for 13 seconds against the wind (hence [tex]\( v - w \)[/tex] for the effective speed). The total distance covered is 52 yards.
2. Second Equation Interpretation:
[tex]\[ 12(v+w)=72 \][/tex]
This equation signifies that Andy flew the airplane for 12 seconds with the wind (hence [tex]\( v + w \)[/tex] for the effective speed). The total distance covered is 72 yards.
Now let's compare these observations with the given options:
- Option A: Andy flew his model airplane at full speed straight into the wind for 12 seconds before turning it around and flying directly with the wind for 13 seconds. The airplane flew 52 yards on the way out and 72 yards on the way back.
- 12 seconds for the way out (against the wind): this does not fit the first equation (which is for 13 seconds).
- 13 seconds for the way back (with the wind): this does not fit the second equation (which is for 12 seconds).
So, Option A does not fit.
- Option B: Andy flew his model airplane at full speed straight into the wind for 13 seconds before turning it around and flying directly with the wind for 12 seconds. The airplane flew 72 yards on the way out and 52 yards on the way back.
- 13 seconds for the way out (against the wind): correct.
- 12 seconds for the way back (with the wind): correct.
- But the distances are reversed. This does not match the distances given in the equations.
So, Option B does not fit either.
- Option C: Andy flew his model airplane at full speed straight into the wind for 13 seconds before turning it around and flying directly with the wind for 12 seconds. The airplane flew 52 yards on the way out and 72 yards on the way back.
- 13 seconds for the way out (against the wind): correct (matches [tex]\(13(v-w)=52\)[/tex]).
- 12 seconds for the way back (with the wind): correct (matches [tex]\(12(v+w)=72\)[/tex]).
- The distances of 52 yards on the way out and 72 yards on the way back are also correct.
Therefore, Option C correctly fits the given system of equations.
- Option D: Andy flew his model airplane at full speed straight into the wind for 12 seconds before turning it around and flying directly with the wind for 13 seconds. The airplane flew 72 yards on the way out and 52 yards on the way back.
- 12 seconds for the way out (against the wind): does not fit the first equation (which is for 13 seconds).
- 13 seconds for the way back (with the wind): does not fit the second equation (which is for 12 seconds).
So, Option D does not fit either.
Hence, the correct situation that corresponds to the system of equations is:
C. Andy flew his model airplane at full speed straight into the wind for 13 seconds before turning it around and flying directly with the wind for 12 seconds. The airplane flew 52 yards on the way out and 72 yards on the way back.
The system of equations:
[tex]\[ \left\{\begin{array}{l} 13(v-w)=52 \\ 12(v+w)=72 \end{array}\right. \][/tex]
Here, [tex]\( v \)[/tex] represents the speed of the airplane in still air, and [tex]\( w \)[/tex] represents the speed of the wind.
1. First Equation Interpretation:
[tex]\[ 13(v-w)=52 \][/tex]
This equation signifies that Andy flew the airplane for 13 seconds against the wind (hence [tex]\( v - w \)[/tex] for the effective speed). The total distance covered is 52 yards.
2. Second Equation Interpretation:
[tex]\[ 12(v+w)=72 \][/tex]
This equation signifies that Andy flew the airplane for 12 seconds with the wind (hence [tex]\( v + w \)[/tex] for the effective speed). The total distance covered is 72 yards.
Now let's compare these observations with the given options:
- Option A: Andy flew his model airplane at full speed straight into the wind for 12 seconds before turning it around and flying directly with the wind for 13 seconds. The airplane flew 52 yards on the way out and 72 yards on the way back.
- 12 seconds for the way out (against the wind): this does not fit the first equation (which is for 13 seconds).
- 13 seconds for the way back (with the wind): this does not fit the second equation (which is for 12 seconds).
So, Option A does not fit.
- Option B: Andy flew his model airplane at full speed straight into the wind for 13 seconds before turning it around and flying directly with the wind for 12 seconds. The airplane flew 72 yards on the way out and 52 yards on the way back.
- 13 seconds for the way out (against the wind): correct.
- 12 seconds for the way back (with the wind): correct.
- But the distances are reversed. This does not match the distances given in the equations.
So, Option B does not fit either.
- Option C: Andy flew his model airplane at full speed straight into the wind for 13 seconds before turning it around and flying directly with the wind for 12 seconds. The airplane flew 52 yards on the way out and 72 yards on the way back.
- 13 seconds for the way out (against the wind): correct (matches [tex]\(13(v-w)=52\)[/tex]).
- 12 seconds for the way back (with the wind): correct (matches [tex]\(12(v+w)=72\)[/tex]).
- The distances of 52 yards on the way out and 72 yards on the way back are also correct.
Therefore, Option C correctly fits the given system of equations.
- Option D: Andy flew his model airplane at full speed straight into the wind for 12 seconds before turning it around and flying directly with the wind for 13 seconds. The airplane flew 72 yards on the way out and 52 yards on the way back.
- 12 seconds for the way out (against the wind): does not fit the first equation (which is for 13 seconds).
- 13 seconds for the way back (with the wind): does not fit the second equation (which is for 12 seconds).
So, Option D does not fit either.
Hence, the correct situation that corresponds to the system of equations is:
C. Andy flew his model airplane at full speed straight into the wind for 13 seconds before turning it around and flying directly with the wind for 12 seconds. The airplane flew 52 yards on the way out and 72 yards on the way back.
