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 telling $t$, the time in minutes represent the independent variable and $k$, the number of kilometers, represent the dependent variable.
Which equation can be used to represent $k$, the number of kilometers Julissa runs in $t$ minutes?
A. $k-2=\frac{1}{9}(t-18)$
B. $k-18=\frac{1}{9}(t-2)$
C. $k-2=9(t-18)$
D. $k-18=9(t-2)$
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 of a linear equation: k − k 1 = m ( t − t 1 ) .
Substitute the point (18, 2) and the slope into the equation: k − 2 = 9 1 ( t − 18 ) .
The equation representing the relationship between k and t is: k − 2 = 9 1 ( t − 18 ) .
Explanation
Understanding the Problem Let's analyze the problem. We are given two points (18, 2) and (54, 6) that represent the time in minutes and the corresponding distance in kilometers Julissa has run. We need to find the equation that represents the relationship between time t and distance k . Since Julissa is running at a constant pace, this relationship is linear.
Calculating the Slope First, we need to find the slope of the line. The slope, m , is given by the formula: m = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the two points. In this case, ( x 1 , y 1 ) = ( 18 , 2 ) and ( x 2 , y 2 ) = ( 54 , 6 ) . Substituting these values into the formula, we get: m = 54 − 18 6 − 2 = 36 4 = 9 1
Finding the Equation Now that we have the slope, we can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is given by: y − y 1 = m ( x − x 1 ) In our case, this translates to: k − k 1 = m ( t − t 1 ) We can use the point (18, 2) as ( t 1 , k 1 ) . Substituting the values, we get: k − 2 = 9 1 ( t − 18 )
Matching the 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 amount of fuel in your car as you drive, you can use a linear equation to predict how much fuel you'll have left after a certain distance. Similarly, businesses use linear models to predict costs, revenue, and profits based on different levels of production or sales. This problem demonstrates how to create a linear model from given data points, which is a fundamental skill in data analysis and decision-making.
The equation that represents the relationship between the time Julissa runs and the kilometers she covers is k − 2 = 9 1 ( t − 18 ) . This equation is derived from the slope calculated using the provided data points. The correct answer is option A.
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