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).
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.
To start a dog walking business and analyze the given data, we can use the concept of linear relationships and create equations based on the given information.
For the dog walking business:
Let's assume that the cost of the dog-walking service is directly proportional to the number of dogs walked. We can represent this relationship using the equation: y = mx.
Given data:
(2, $30) implies that when 2 dogs are walked, it costs $30.
(3, $45) implies that when 3 dogs are walked, it costs $45.
(5, $75) implies that when 5 dogs are walked, it costs $75.
(7, $105) implies that when 7 dogs are walked, it costs $105.
We can use the first two data points (2, $30) and (3, $45) to find the value of m in the equation y = mx.
Substituting the first data point (2, $30):
30 = 2m
Substituting the second data point (3, $45):
45 = 3m
Now, solve these two equations simultaneously:
30 = 2m
45 = 3m
Divide the second equation by 3:
45/3 = m
15 = m
Therefore, the equation representing the relationship between the number of dogs walked (x) and the cost of the service (y) is:
y = 15x
Now, let's analyze the data for driving to the hotel:
Let's assume that the time it takes to reach the hotel is inversely proportional to the speed (x) while driving. We can represent this relationship using the equation: y = k/x.
Given data:
(55 mph, 109 minutes) implies that when driving at 55 mph, it takes 109 minutes.
(35 mph, 171 minutes) implies that when driving at 35 mph, it takes 171 minutes.
(60 mph, 100 minutes) implies that when driving at 60 mph, it takes 100 minutes.
(80 mph, 75 minutes) implies that when driving at 80 mph, it takes 75 minutes.
We can use the first two data points (55 mph, 109 minutes) and (35 mph, 171 minutes) to find the value of k in the equation y = k/x.
Substituting the first data point (55 mph, 109 minutes):
109 = k/55
Substituting the second data point (35 mph, 171 minutes):
171 = k/35
Now, solve these two equations simultaneously:
109 = k/55
171 = k/35
Multiply the first equation by 55:
109 * 55 = k
Multiply the second equation by 35:
171 * 35 = k
Therefore, k = 5995.
The equation representing the relationship between the speed (x) and the time it takes to reach the hotel (y) is:
y = 5995/x
Please note that these equations are approximate representations of the given data points, and should be used as a reference for further analysis or predictions.