The table shows the number of cinnamon rolls remaining each hour after opening at a local coffee shop.
1 226
3 178
6 106
10 10
Based on the table, which equation models this situation?
The table shows that the number of cinnamon rolls remaining decreases over time. Since this is a decreasing pattern, we can model it with a linear equation.
The equation that models this situation is a linear equation of the form y = mx + b, where y represents the number of cinnamon rolls remaining and x represents the number of hours after opening.
To determine the equation, we can use the values in the table to find the slope (m) and y-intercept (b) of the equation.
Using the points (1, 226) and (10, 10):
Slope (m) = (y2 - y1) / (x2 - x1)
m = (10 - 226) / (10 - 1)
m = -216 / 9
m = -24
Now, we can substitute the values of one of the points (1, 226) into the equation to find the y-intercept (b):
226 = (-24)(1) + b
226 = -24 + b
b = 226 + 24
b = 250
Therefore, the equation that models this situation is:
y = -24x + 250
To model this situation, we need to find an equation that relates the number of cinnamon rolls remaining to the number of hours since the coffee shop has opened.
Let's start by looking at the pattern in the data:
- After 1 hour, there are 226 cinnamon rolls remaining.
- After 3 hours, there are 178 cinnamon rolls remaining.
- After 6 hours, there are 106 cinnamon rolls remaining.
- After 10 hours, there are 10 cinnamon rolls remaining.
From the data, it appears that the number of cinnamon rolls remaining is decreasing over time.
To find the equation, we can use the concept of linear regression. We can assume that the relationship between the number of cinnamon rolls remaining and the number of hours is a linear one.
Let's define our variables:
- Let x represent the number of hours since the coffee shop has opened.
- Let y represent the number of cinnamon rolls remaining.
Now, let's find the slope of the line using the formula:
slope = (y2 - y1) / (x2 - x1)
slope = (178 - 226) / (3 - 1) = -48 / 2 = -24
Next, let's find the y-intercept of the line by substituting the values of x and y into the equation y = mx + b, where m is the slope:
226 = -24(1) + b
226 = -24 + b
Now, solve for b:
b = 226 + 24 = 250
Therefore, our equation is y = -24x + 250.
This equation models the situation described in the table.
To determine which equation models the situation, we need to analyze the relationship between the number of cinnamon rolls remaining and the time in hours.
First, let's identify the pattern:
From the given data, we can observe that the number of cinnamon rolls is decreasing over time. As time progresses, the number of cinnamon rolls decreases.
Now, let's express this pattern using an equation:
Let's consider that "x" represents the time elapsed in hours, and "y" represents the number of cinnamon rolls remaining.
By observing the data, we can see that the number of cinnamon rolls remaining is decreasing linearly over time. This indicates that there is a constant rate of change.
To determine this, we can calculate the difference in the number of cinnamon rolls between each given data point:
From hour 1 to hour 3: 226 - 178 = 48
From hour 3 to hour 6: 178 - 106 = 72
From hour 6 to hour 10: 106 - 10 = 96
By evaluating these differences, we can find a pattern. Each subsequent difference is twice as large as the previous difference.
Therefore, the constant rate of change is -48 (negative because the number of cinnamon rolls is decreasing by that amount each hour).
Now, let's write the equation in slope-intercept form (y = mx + b), where "m" is the slope (constant rate of change) and "b" is the y-intercept (initial number of cinnamon rolls):
Using the point (1, 226) where "x = 1" and "y = 226":
226 = (-48)(1) + b
226 = -48 + b
b = 274
Now, we can write the equation:
y = -48x + 274
Therefore, the equation that models this situation is y = -48x + 274.