Dennis has a problem. He went to school, which is 12 blocks @60 degrees celcius away from home. then he went to the intramural scrimmage at Dumbbell Hight football field, 14 blocks @ (7pie/4)from school. Then he had a study date with aula, and her house is -12 blocks x from the football field. The library they studied at is 11 blocks Y from Paula's. Aftger studying, the went to go see a romantic comedy at the movie theatre, -8.5x - 8.5y from the library. paula's Dad is the protective kind, so her parents picked Paula up at the theatre. Dennis will run out of gas 5 blocks away from the theatre according to his hightech gas gauge. If that happens, can Dennis make it home on these weird city streets (assume Dennis can travel in any direction)? If yes, what direction does Dennis need to travel in from the movies? If no, how close can he get to home?

Note: there are (2pie/360 degrees).

I assume the @60 degrees celcius is a typo. Add the X components of all travel, then the Y components. That is the final position. Reverse the direction to get home.

To solve the problem, let's break it down step by step:

1. Distance from home to school: 12 blocks @60 degrees Celsius away from home. Since we assume the 60 degrees Celsius is a typo, we'll ignore it for now. The direction is not specified, so we'll consider it as a straight line from home to school.

2. Distance from school to Dumbbell Hight football field: 14 blocks @(7π/4). Again, the direction is not specified, so we'll consider it as a straight line.

3. Distance from Dumbbell Hight football field to Paula's house: -12 blocks x from the football field. The negative sign indicates that this distance is in the opposite direction from the football field.

4. Distance from Paula's house to the library: 11 blocks Y from Paula's. The specific direction is not given, so we'll assume it is a straight line.

5. Distance from the library to the movie theatre: -8.5x - 8.5y from the library. The negative signs indicate that this distance is in the opposite direction from the library in both the x- and y-axis.

Now, let's add up the x and y components of Dennis' travel:

X components:
- Distance from home to school: 12 blocks (assume x = 12)
- Distance from school to football field: No x-component mentioned, so we'll assume it to be 0.
- Distance from football field to Paula's: -12 blocks (assume x = -12)
- Distance from library to theatre: -8.5x (assume x = -8.5)

Y components:
- Distance from school to football field: No y-component mentioned, so we'll assume it to be 0.
- Distance from Paula's to library: 11 blocks (assume y = 11)
- Distance from library to theatre: -8.5y (assume y = -8.5)

Calculating the final position:
X component: 12 - 12 - 8.5*(-8.5) = 12 - 12 + 72.25 = 72.25 blocks (assumed positive direction is towards Dennis' home)

Y component: 0 + 11 + 8.5*(-8.5) = 0 + 11 - 72.25 = -61.25 blocks (assumed negative direction is away from Dennis' home)

Therefore, Dennis' final position is approximately (72.25, -61.25) blocks.

Based on the position, Dennis is 72.25 blocks away from home in the x-direction and 61.25 blocks away from home in the y-direction. To get back home, Dennis needs to reverse the direction of his travel, so he needs to travel in the opposite direction of the x and y components.