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 :

  • We identified the temperature equations: \(T_{KC}(h) = -2 - 2h\) and \(T_{CH}(h) = 6 - 4h\).
  • We set the equations equal to each other: \(-2 - 2h = 6 - 4h\).
  • We solved for \(h\) by isolating the variable on one side and the constant terms on the other, finding \(h = 4\).
  • The correct equation is \(-2 - 2h = 6 - 4h\) and the time taken is \(\boxed{4}\) hours.

Explanation

  1. Problem Analysis and Equation Identification

We are given the initial temperatures and the rates of change for two cities:

  • Kansas City: Starting temperature \(T_{KC} = -2^{\circ}\text{F}\), rate of change \(-2^{\circ}\text{F}\) per hour. The temperature after \(h\) hours is \(T_{KC}(h) = -2 - 2h\).
  • Chicago: Starting temperature \(T_{CH} = 6^{\circ}\text{F}\), rate of change \(-4^{\circ}\text{F}\) per hour. The temperature after \(h\) hours is \(T_{CH}(h) = 6 - 4h\).

We need to find the number of hours \(h\) such that the temperatures in both cities are equal. This occurs when \(T_{KC}(h) = T_{CH}(h)\), which gives us the equation \(-2 - 2h = 6 - 4h\).

  1. Solving for the Number of Hours

To find the time \(h\) when the temperatures reach the same value, we solve the equation:
\[-2 - 2h = 6 - 4h\]

First, add \(4h\) to both sides of the equation:
\[-2 - 2h + 4h = 6\]
\[-2 + 2h = 6\]

Next, add \(2\) to both sides of the equation to isolate the term with \(h\):
\[2h = 6 + 2\]
\[2h = 8\]

Finally, divide both sides by \(2\):
\[h = \frac{8}{2}\]
\[h = 4\]
So, it will take \(4\) hours for the temperatures in Kansas City and Chicago to be the same.

Examples

This type of problem, where two rates of change are set equal to each other, is foundational in many real-world applications. For instance, if two companies offer different subscription models—one with a high initial fee but low monthly rate, and another with a low initial fee but high monthly rate—you can use this method to determine at which month the total cost of both subscriptions will be exactly the same. It helps in making informed financial decisions by identifying the break-even point in time for different cost structures.

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