Kaya is hiking down and Katie is hiking up.
a) Determine the algebraic equation of each graph relating y= elevation (ft) to x = # hours.
Katie’s hike: Starts at 1600 ft and ends at 5100 feet.
Kaya’s hike: Starts at 7000 ft ends at 3000 ft.
The graph goes from 0-5 hours and 0-8000 feet.
To create the equations for each lady's hike, we must first know the slope of the line, assuming these are constant rates of increase/decrease. Since the graphs go from 0 through 5 hours, our change in x-values is 5, and our change in y-values is the difference in height:
Katie: Slope = (y2-y1)/(x2-x1) = (5100-1600)/(5-0) = 700 (she increases 700 feet in height per hour)
Kaya: Slope = (y2-y1)/(x2-x1) = (3000-7000)/(5-0) = -800 (she decreases 800 feet in height per hour)
To create the equation, we can easily use the slope-intercept form of a linear equation because the initial values (at time x = 0) ARE the y-intercepts of the graph. Therefore:
Katie's Equation: y = 700x + 1600
Kaya's Equation: y = -800x + 7000
b) Solve algebraically to determine the exact time and elevation when Kaya and Katie are at the same elevation.
Here, we simply take Katie's equation and set it equal to Kaya's equation and solve for the value of x:
700x + 1600 = -800x + 7000 [Adding 800x and subtracting 1600 from both sides...]
1500x = 5400 [Dividing 1500 from both sides...]
x = 3.6
Therefore, Katie and Kaya will be at the same height 3.6 hours (or 3 hours and 36 minutes) into their respective hikes (assuming they began moving at the same time and both moved at constant rates).
