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°F and will drop by 4°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 :

  • The correct equation to model the scenario is \(-2 - 2h = 6 - 4h\).
  • To solve, add \(4h\) to both sides and add \(2\) to both sides, resulting in \(2h = 8\).
  • Dividing by \(2\) gives \(h = 4\).
  • Final answer: \(\boxed{4}\) hours.

Explanation

  1. Modeling the Temperature Functions

To determine when the temperatures in Kansas City and Chicago will be equal, we first model the temperature in each city as a function of time \(h\) (in hours).

For Kansas City, the starting temperature is \(-2^{\circ}\text{F}\) and it drops by \(2^{\circ}\text{F}\) each hour. Therefore, the temperature \(T_{KC}\) at time \(h\) is given by:
\[T_{KC}(h) = -2 - 2h\]

For Chicago, the starting temperature is \(6^{\circ}\text{F}\) and it drops by \(4^{\circ}\text{F}\) each hour. Thus, the temperature \(T_{CHI}\) at time \(h\) is:
\[T_{CHI}(h) = 6 - 4h\]

To find when the temperatures are equal, we set the two expressions equal to each other:
\[-2 - 2h = 6 - 4h\]
Comparing this to the provided options, we see that the first equation, \(-2 - 2h = 6 - 4h\), is the correct model for this scenario.

  1. Solving for Time \(h\)

Now we solve the equation \(-2 - 2h = 6 - 4h\) for \(h\).

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

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

Finally, divide both sides by \(2\):
\[h = \frac{8}{2}\]
\[h = 4\]

Thus, it will take \(4\) hours for the temperatures in the two cities to be equal.

Examples

Understanding how to model changing quantities with linear equations is a fundamental skill in algebra. This concept is useful in many real-life situations, such as comparing the costs of different service plans that change over time, predicting when two vehicles traveling at different speeds will meet, or analyzing trends in data where multiple variables are changing at constant rates. Mastering this technique helps in making informed decisions by accurately calculating the point at which two dynamic conditions intersect.

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