To determine which equation accurately represents the temperature, [tex]\( t \)[/tex], in Burrtown at hour [tex]\( h \)[/tex], let's analyze each option step-by-step based on the given information.
### Given Information:
- The initial temperature at midnight ([tex]\( h = 0 \)[/tex]) is [tex]\( 32^\circ \)[/tex]F.
- For every hour after midnight, the temperature drops by [tex]\( 2^\circ \)[/tex].
### Equation Analysis:
#### Option A: [tex]\( t = 2h + 32 \)[/tex]
- Initial Condition (when [tex]\( h = 0 \)[/tex]):
[tex]\( t = 2(0) + 32 \Rightarrow t = 32^\circ \)[/tex]F.
- Temperature Drop: When [tex]\( h = 1 \)[/tex], [tex]\( t = 2(1) + 32 = 34^\circ \)[/tex]F. This is an increase, not a drop.
Since the temperature rises with increasing [tex]\( h \)[/tex], this equation is not correct.
#### Option B: [tex]\( t = -2h + 32 \)[/tex]
- Initial Condition (when [tex]\( h = 0 \)[/tex]):
[tex]\( t = -2(0) + 32 \Rightarrow t = 32^\circ \)[/tex]F.
- Temperature Drop: When [tex]\( h = 1 \)[/tex], [tex]\( t = -2(1) + 32 = 30^\circ \)[/tex]F. This is a drop of [tex]\( 2^\circ \)[/tex], which matches the given information.
This equation decreases correctly by [tex]\( 2^\circ \)[/tex] per hour, hence it is a valid model of the temperature change.
#### Option C: [tex]\( t = 32h - 2 \)[/tex]
- Initial Condition (when [tex]\( h = 0 \)[/tex]):
[tex]\( t = 32(0) - 2 \Rightarrow t = -2 \)[/tex]F. This does not match the initial temperature of [tex]\( 32^\circ \)[/tex]F at midnight.
Hence, this equation is incorrect.
#### Option D: [tex]\( t = -32h + 2 \)[/tex]
- Initial Condition (when [tex]\( h = 0 \)[/tex]):
[tex]\( t = -32(0) + 2 \Rightarrow t = 2 \)[/tex]F. This does not match the initial temperature of [tex]\( 32^\circ \)[/tex]F at midnight.
Therefore, this equation is incorrect.
### Conclusion
Based on the given conditions and evaluating each equation:
The correct equation representing the temperature [tex]\( t \)[/tex] at hour [tex]\( h \)[/tex] is:
[tex]\[ \boxed{t = -2h + 32} \][/tex]
Thus, option (B) is the correct answer.
