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A meteorologist recorded the temperatures at two weather stations. At noon, the temperature at Station A was 58 degrees Fahrenheit, and it decreased at a constant rate of 5 degrees per hour over the course of the day. At noon, the temperature at Station B was -2 degrees Fahrenheit, and increased at a constant rate of 3 degrees per hour. After how many hours will the two stations have exactly the same temperature?

We can use a table below to record the temperatures at each station after given times. The first row has been completed for you: we can think of the temperatures at noon as 0 hours since that’s our starting temperature. Use the expressions for each station to find their temperatures after each of these times. You can use the calculator to help you solve.
Station A's temperature after t hours:
58+(−5t)
Station B's temperature after t hours:
−2+3t
Hours Station A's
temperature Station B's
temperature
0 58 -2
4
8
12

1 answer

To determine the temperatures at Station A and Station B after each of the specified hours (4, 8, and 12), we will use the given expressions for each station's temperature after \( t \) hours.

Temperatures:

Station A: \( T_A(t) = 58 - 5t \)
Station B: \( T_B(t) = -2 + 3t \)

Now, let's calculate the temperatures for 4, 8, and 12 hours:

At 4 Hours (t = 4):

  • Station A:
    \[ T_A(4) = 58 - 5(4) = 58 - 20 = 38 \]

  • Station B:
    \[ T_B(4) = -2 + 3(4) = -2 + 12 = 10 \]

At 8 Hours (t = 8):

  • Station A:
    \[ T_A(8) = 58 - 5(8) = 58 - 40 = 18 \]

  • Station B:
    \[ T_B(8) = -2 + 3(8) = -2 + 24 = 22 \]

At 12 Hours (t = 12):

  • Station A:
    \[ T_A(12) = 58 - 5(12) = 58 - 60 = -2 \]

  • Station B:
    \[ T_B(12) = -2 + 3(12) = -2 + 36 = 34 \]

Summary Table:

Let's fill out the temperature values for the hours:

| Hours | Station A's Temperature | Station B's Temperature | |-------|-------------------------|-------------------------| | 0 | 58 | -2 | | 4 | 38 | 10 | | 8 | 18 | 22 | | 12 | -2 | 34 |

Finding the Intersection:

Now, to determine the time when the temperatures are the same, we can set the equations equal to each other:

\[ 58 - 5t = -2 + 3t \]

Now, solve for \( t \):

\[ 58 + 2 = 3t + 5t \ 60 = 8t \ t = \frac{60}{8} = 7.5 \]

Thus, the two stations will have the same temperature after 7.5 hours.