A meteorologist recorded the temperatures at two weather stations in Montana. At midnight, the temperature at Station A was -8 degrees Fahrenheit, and it increased at a constant rate of 3 degrees per hour. At midnight, the temperature at Station B was -2 degrees Fahrenheit, and increased at a constant rate of 2 degrees per hour. After how many hours will the two stations have exactly the same temperature?

Question

Write an expression that represents the temperature at Station A. Let x represent the number of hours since midnight.
Station A
temperature at midnight: −8 degrees
increasing 3 degrees per hour

Answer 1

To write an expression for the temperature at Station A, we start with the initial temperature and then add the increase in temperature over time based on the number of hours since midnight (represented by \( x \)).

1. The initial temperature at Station A is -8 degrees Fahrenheit.
2. The temperature increases at a rate of 3 degrees per hour.

The expression for the temperature at Station A \( T_A \) after \( x \) hours is: \[ T_A = -8 + 3x \]

Next, we can write a similar expression for the temperature at Station B.

1. The initial temperature at Station B is -2 degrees Fahrenheit.
2. The temperature increases at a rate of 2 degrees per hour.

The expression for the temperature at Station B \( T_B \) after \( x \) hours is: \[ T_B = -2 + 2x \]

To find when the temperatures at the two stations are equal, we set the two expressions equal to each other: \[ -8 + 3x = -2 + 2x \]

Now, we can solve for \( x \):

1. Subtract \( 2x \) from both sides: \[ -8 + 3x - 2x = -2 \] \[ -8 + x = -2 \]

2. Add 8 to both sides: \[ x = -2 + 8 \] \[ x = 6 \]

The two stations will have exactly the same temperature after 6 hours.

To summarize:

• The temperature at Station A is represented by \( T_A = -8 + 3x \).
• The temperature at Station B is represented by \( T_B = -2 + 2x \).
• The two stations will have the same temperature after 6 hours.