Julissa is running a 10-kilometer race at a constant pace. After running for 18 minutes, she completes 2 kilometers. After running for 54 minutes, she completes 6 kilometers. Her trainer writes an equation letting [tex]$t$[/tex], the time in minutes, represent the independent variable and [tex]$k$[/tex], the number of kilometers, represent the dependent variable.
Which equation can be used to represent [tex]$k$[/tex], the number of kilometers Julissa runs in [tex]$t$[/tex] minutes?
[tex]$k-2=\frac{1}{9}(t-18)$[/tex]
[tex]$k-18=\frac{1}{9}(t-2)$[/tex]
[tex]$k-2=9(t-18)$[/tex]
[tex]$k-18=9(t-2)$[/tex]
Answer (2)
Calculate the slope using the points (18, 2) and (54, 6): m = 54 − 18 6 − 2 = 9 1 .
Use the point-slope form with the point (18, 2): k − 2 = 9 1 ( t − 18 ) .
The equation representing the relationship is k − 2 = 9 1 ( t − 18 ) .
Therefore, the answer is k − 2 = 9 1 ( t − 18 ) .
Explanation
Understanding the Problem We are given two points representing Julissa's progress in the race: (18 minutes, 2 kilometers) and (54 minutes, 6 kilometers). We need to find the equation that represents the relationship between the time t (in minutes) and the distance k (in kilometers). Since Julissa is running at a constant pace, this relationship will be linear.
Calculating the Slope First, we need to find the slope of the line. The slope m is given by the formula: m = t 2 − t 1 k 2 − k 1 Using the given points (18, 2) and (54, 6), we have: m = 54 − 18 6 − 2 = 36 4 = 9 1
Using Point-Slope Form Now that we have the slope, we can use the point-slope form of a linear equation, which is: k − k 1 = m ( t − t 1 ) We can use either of the given points. Let's use the point (18, 2). Substituting m = 9 1 and the point (18, 2) into the point-slope form, we get: k − 2 = 9 1 ( t − 18 )
Finding the Correct Equation Comparing this equation with the given options, we see that it matches the first option: k − 2 = 9 1 ( t − 18 )
Examples
Understanding linear relationships, like the one in this problem, is useful in many real-world situations. For example, if you are tracking the distance you drive over time on a road trip, you can use a similar equation to predict how far you will travel in a certain amount of time, assuming you maintain a constant speed. This can help you estimate arrival times or plan fuel stops.
The equation that represents the kilometers Julissa runs in relation to time is k − 2 = 9 1 ( t − 18 ) . This was derived by calculating the slope from her running data and using the point-slope form of a linear equation with the point (18, 2). Therefore, the correct answer is the first option provided by the trainer.
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