Question
Emelyn is comparing the rates of two different gardening companies for her camp site. The table below shows the rates for Log Edge Landscaping. Log Edge Landscaping Hours Worked 1.5 3 3.5 4.5 Total Cost $26 $44 $50 $62 The equation below represents the rates for Gatewood Gardening Services, where x represents the hours worked and y represents the total cost, in dollars. Compare the rates of both companies, and graph the relationship that represents the cost for the company with a greater rate of change.
Answers
GPT-4o mini
To compare the rates of the two gardening companies, we first need to analyze the data provided for Log Edge Landscaping and calculate the rate of change (slope) based on the hours worked and the corresponding costs.
### Log Edge Landscaping Data:
- **Hours Worked (x)**: 1.5, 3, 3.5, 4.5
- **Total Cost (y)**: $26, $44, $50, $62
We can calculate the rate of change (slope) between various points using the formula:
\[
\text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let's calculate the slope between the first two points (1.5 hours to 3 hours):
\[
\text{slope} = \frac{44 - 26}{3 - 1.5} = \frac{18}{1.5} = 12
\]
Next, let’s calculate the slope between the second pair (3 hours to 3.5 hours):
\[
\text{slope} = \frac{50 - 44}{3.5 - 3} = \frac{6}{0.5} = 12
\]
Now, for the last pair (3.5 hours to 4.5 hours):
\[
\text{slope} = \frac{62 - 50}{4.5 - 3.5} = \frac{12}{1} = 12
\]
The rate of change for Log Edge Landscaping is consistently $12 per hour.
### Gatewood Gardening Services:
For Gatewood Gardening Services, we have an equation given for the cost:
\[
y = mx + b
\]
where \(m\) is the rate (slope) and \(b\) is the y-intercept. If no specific values for \(m\) and \(b\) are provided in your question, we will need to know the slope or an example of costing to do a comparison.
Assuming an example based on typical data, let’s consider that the cost function for Gatewood Gardening Services has a slope of $15 per hour (please adjust according to actual data you have):
### Comparison of Rates:
- **Log Edge Landscaping**: $12 per hour
- **Gatewood Gardening Services**: $15 per hour
### Conclusion:
Since the rate of change (cost per hour) for Gatewood Gardening Services is greater, we can conclude that Gatewood Gardening Services has a greater rate.
### Graphing:
To graph these two companies' rates, you can plot the following data:
- For **Log Edge Landscaping**, the points would be:
- (1.5, 26)
- (3, 44)
- (3.5, 50)
- (4.5, 62)
- For **Gatewood Gardening Services** (assuming it starts at $0 and has a cost of $15 per hour, the two points could be):
- (0, 0)
- (4, 60) (4 hours of work at $15/hour)
On a graph:
- The x-axis would represent hours worked.
- The y-axis would represent total cost.
Plot these points and draw lines connecting them:
1. For Log Edge, draw a line through the points representing consistent increases in cost.
2. For Gatewood, draw a line from the origin with a steeper slope (if based on $15/hour).
This visual representation will help Emelyn clearly see which company has a higher cost based on hours worked.
### Log Edge Landscaping Data:
- **Hours Worked (x)**: 1.5, 3, 3.5, 4.5
- **Total Cost (y)**: $26, $44, $50, $62
We can calculate the rate of change (slope) between various points using the formula:
\[
\text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let's calculate the slope between the first two points (1.5 hours to 3 hours):
\[
\text{slope} = \frac{44 - 26}{3 - 1.5} = \frac{18}{1.5} = 12
\]
Next, let’s calculate the slope between the second pair (3 hours to 3.5 hours):
\[
\text{slope} = \frac{50 - 44}{3.5 - 3} = \frac{6}{0.5} = 12
\]
Now, for the last pair (3.5 hours to 4.5 hours):
\[
\text{slope} = \frac{62 - 50}{4.5 - 3.5} = \frac{12}{1} = 12
\]
The rate of change for Log Edge Landscaping is consistently $12 per hour.
### Gatewood Gardening Services:
For Gatewood Gardening Services, we have an equation given for the cost:
\[
y = mx + b
\]
where \(m\) is the rate (slope) and \(b\) is the y-intercept. If no specific values for \(m\) and \(b\) are provided in your question, we will need to know the slope or an example of costing to do a comparison.
Assuming an example based on typical data, let’s consider that the cost function for Gatewood Gardening Services has a slope of $15 per hour (please adjust according to actual data you have):
### Comparison of Rates:
- **Log Edge Landscaping**: $12 per hour
- **Gatewood Gardening Services**: $15 per hour
### Conclusion:
Since the rate of change (cost per hour) for Gatewood Gardening Services is greater, we can conclude that Gatewood Gardening Services has a greater rate.
### Graphing:
To graph these two companies' rates, you can plot the following data:
- For **Log Edge Landscaping**, the points would be:
- (1.5, 26)
- (3, 44)
- (3.5, 50)
- (4.5, 62)
- For **Gatewood Gardening Services** (assuming it starts at $0 and has a cost of $15 per hour, the two points could be):
- (0, 0)
- (4, 60) (4 hours of work at $15/hour)
On a graph:
- The x-axis would represent hours worked.
- The y-axis would represent total cost.
Plot these points and draw lines connecting them:
1. For Log Edge, draw a line through the points representing consistent increases in cost.
2. For Gatewood, draw a line from the origin with a steeper slope (if based on $15/hour).
This visual representation will help Emelyn clearly see which company has a higher cost based on hours worked.