For the dog walking business:
- The slope of the line (the rate at which the cost increases per dog) is (y2 - y1) / (x2 - x1), which can be calculated using any two of the given points. Using (2, $30) and (5, $75): (75 - 30) / (5 - 2) = $15 per dog.
- The y-intercept of the line (the starting cost, even if no dogs are walked) is the value of y when x = 0. Since this doesn't make sense for a dog walking business, we can't calculate it from the given data.
- The equation for the line (how much it would cost to walk any number of dogs) is y = mx + b, where m is the slope and b is the y-intercept (which we don't have).
For the car trip:
- The slope of the line (how much longer the trip takes for every 1 mph slower I drive) is (y2 - y1) / (x2 - x1), which can be calculated using any two of the given points. Using (55 mph, 109 minutes) and (60 mph, 100 minutes): (100 - 109) / (60 - 55) = -2 minutes per mph (the negative sign indicates that the time decreases as the speed increases).
- The y-intercept of the line (the time it would take to make the trip at 0 mph) is the value of y when x = 0. This doesn't make sense for a car trip, so we can't calculate it from the given data.
- The equation for the line (how long the trip would take at any speed) is y = mx + b, where m is the slope and b is the y-intercept (which we don't have). Note that while the slope is negative, the time (y) cannot be negative, so the line will level off at some minimum value.
I decide to start a dog walking business, x is the amount of dogs walked and y is how much it costs. The data for my first four days is (2, $30) (3, $45), (5, $75), and (7, $105). I am driving to a hotel 100 miles away, as traffic decreases I can drive faster and when it increases I have to slow down, y is how many minutes it will take to get there, x is my speed in mph. The data is (55 mph, 109 minutes), (35 mph, 171 minutes), (60 mph, 100 minutes), and (80 mph, 75 minutes).
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