Answered

On a winter evening at sundown, a Kansas City news station reports that the temperature is -2°F and will drop by 2°F per hour until sunrise. At the same time, a Chicago news station reports that the temperature is $6^{\circ} F$ and will drop by $4^{\circ} F$ per hour.



Which equation can you use to find $h$, the number of hours it will take the cities to reach the same temperature?

A$-2 - 2h = 6 - 4h$
B$2h - 2 = 6h - 4$



How long will it take for the cities to reach the same temperature?

Simplify any fractions.

____ hours

Verified Answer :

  • Define the temperature functions for both cities: \(T_K(h) = -2 - 2h\) and \(T_C(h) = 6 - 4h\).
  • Equate the two expressions: \(-2 - 2h = 6 - 4h\).
  • Solve the equation by isolating \(h\), which gives \(2h = 8\), so \(h = 4\).
  • The time taken for the cities to reach the same temperature is \(\boxed{4}\) hours.

Explanation

  1. Setting up the Equation

The problem describes two cities with different starting temperatures and different rates of temperature drop. We are tasked with finding the number of hours \(h\) until these two cities reach the same temperature.

For Kansas City, the initial temperature is \(-2^{\circ}\text{F}\) and it drops at a rate of \(2^{\circ}\text{F}\) per hour. The temperature at any time \(h\) (in hours) can be expressed as:
\[T_K(h) = -2 - 2h\]

For Chicago, the initial temperature is \(6^{\circ}\text{F}\) and it drops at a rate of \(4^{\circ}\text{F}\) per hour. The temperature at any time \(h\) can be expressed as:
\[T_C(h) = 6 - 4h\]

To find the time \(h\) at which the temperatures are equal, we set the two expressions equal to each other:
\[-2 - 2h = 6 - 4h\]
This matches option (a).

  1. Solving for h

To solve for \(h\), we start with the equation:
\[-2 - 2h = 6 - 4h\]
We want to isolate \(h\) on one side of the equation. First, add \(4h\) to both sides:
\[-2 - 2h + 4h = 6\]
\[-2 + 2h = 6\]
Next, add \(2\) to both sides:
\[2h = 6 + 2\]
\[2h = 8\]
Finally, divide by 2:
\[h = \frac{8}{2} = 4\]
It will take 4 hours for the temperatures in Kansas City and Chicago to be equal.

  1. Conclusion

The correct equation is \(-2 - 2h = 6 - 4h\), and it will take 4 hours for the cities to reach the same temperature. The final answer is \(\boxed{4}\) hours.

Examples

This type of problem, where two rates are compared to find an intersection point, is common in resource management and logistics. For instance, if two batteries are discharging at different rates, you could calculate exactly when they will reach the same power level. Understanding linear relationships allows for better planning and monitoring in various technical and real-world systems.

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