Question
Charlotte is deciding between two landscaping companies for her place of business. Company A charges $50 per hour and a $100 equipment fee. Company B charges $25 per hour and a $200 equipment fee. Let AA represent the amount Company A would charge for tt hours of landscaping, and let BB represent the amount Company B would charge for tt hours of landscaping. Graph each function and determine which company would be cheaper if Charlotte needs 7 hours of landscaping.
Answers
GPT-4o mini
To find out which landscaping company is cheaper for Charlotte, we need to write equations for each company based on their charges and then evaluate those equations at \( t = 7 \) hours.
### Company A:
- Charges: $50 per hour
- Equipment fee: $100
The total cost for Company A, represented as \( A(t) \), for \( t \) hours can be expressed with the formula:
\[
A(t) = 50t + 100
\]
### Company B:
- Charges: $25 per hour
- Equipment fee: $200
The total cost for Company B, represented as \( B(t) \), for \( t \) hours is:
\[
B(t) = 25t + 200
\]
### Calculating Costs for 7 Hours:
Now, we will compute the costs for both companies when \( t = 7 \) hours.
1. **For Company A:**
\[
A(7) = 50 \times 7 + 100 = 350 + 100 = 450
\]
2. **For Company B:**
\[
B(7) = 25 \times 7 + 200 = 175 + 200 = 375
\]
### Comparison:
- Total cost for Company A for 7 hours: **$450**
- Total cost for Company B for 7 hours: **$375**
### Conclusion:
Since \( B(7) = 375 < A(7) = 450 \), Company B is the cheaper option for Charlotte if she needs 7 hours of landscaping services.
### Graphing the Functions:
To visualize this, you could graph the functions \( A(t) = 50t + 100 \) and \( B(t) = 25t + 200 \).
- For Company A, the graph is a line with a slope of 50 that starts at 100 on the y-axis.
- For Company B, the graph is a line with a slope of 25 that starts at 200 on the y-axis.
The intersection of these two lines will give the point where both companies charge the same price. Calculating that point involves setting \( A(t) = B(t) \):
\[
50t + 100 = 25t + 200
\]
Solving for \( t \):
\[
50t - 25t = 200 - 100 \\
25t = 100 \\
t = 4
\]
At \( t = 4 \), both companies charge the same, which can be calculated as follows:
\[
A(4) = 50 \times 4 + 100 = 200 + 100 = 300 \\
B(4) = 25 \times 4 + 200 = 100 + 200 = 300
\]
Thus, for any \( t < 4 \), Company B is cheaper and for \( t > 4 \), Company A becomes the cheaper option.
### Company A:
- Charges: $50 per hour
- Equipment fee: $100
The total cost for Company A, represented as \( A(t) \), for \( t \) hours can be expressed with the formula:
\[
A(t) = 50t + 100
\]
### Company B:
- Charges: $25 per hour
- Equipment fee: $200
The total cost for Company B, represented as \( B(t) \), for \( t \) hours is:
\[
B(t) = 25t + 200
\]
### Calculating Costs for 7 Hours:
Now, we will compute the costs for both companies when \( t = 7 \) hours.
1. **For Company A:**
\[
A(7) = 50 \times 7 + 100 = 350 + 100 = 450
\]
2. **For Company B:**
\[
B(7) = 25 \times 7 + 200 = 175 + 200 = 375
\]
### Comparison:
- Total cost for Company A for 7 hours: **$450**
- Total cost for Company B for 7 hours: **$375**
### Conclusion:
Since \( B(7) = 375 < A(7) = 450 \), Company B is the cheaper option for Charlotte if she needs 7 hours of landscaping services.
### Graphing the Functions:
To visualize this, you could graph the functions \( A(t) = 50t + 100 \) and \( B(t) = 25t + 200 \).
- For Company A, the graph is a line with a slope of 50 that starts at 100 on the y-axis.
- For Company B, the graph is a line with a slope of 25 that starts at 200 on the y-axis.
The intersection of these two lines will give the point where both companies charge the same price. Calculating that point involves setting \( A(t) = B(t) \):
\[
50t + 100 = 25t + 200
\]
Solving for \( t \):
\[
50t - 25t = 200 - 100 \\
25t = 100 \\
t = 4
\]
At \( t = 4 \), both companies charge the same, which can be calculated as follows:
\[
A(4) = 50 \times 4 + 100 = 200 + 100 = 300 \\
B(4) = 25 \times 4 + 200 = 100 + 200 = 300
\]
Thus, for any \( t < 4 \), Company B is cheaper and for \( t > 4 \), Company A becomes the cheaper option.
how would that be marked on a graph
GPT-4o mini
To graph the functions for the two landscaping companies, you would plot their total cost equations on a coordinate plane where the x-axis represents the number of hours \( t \) and the y-axis represents the total cost.
### Step-by-step Instructions to Graph \( A(t) \) and \( B(t) \):
1. **Identify the functions:**
- Company A: \( A(t) = 50t + 100 \)
- Company B: \( B(t) = 25t + 200 \)
2. **Determine key points:**
- For \( A(t) \):
- When \( t = 0 \): \( A(0) = 100 \)
- When \( t = 1 \): \( A(1) = 150 \)
- When \( t = 2 \): \( A(2) = 200 \)
- When \( t = 3 \): \( A(3) = 250 \)
- When \( t = 4 \): \( A(4) = 300 \)
- When \( t = 5 \): \( A(5) = 350 \)
- When \( t = 6 \): \( A(6) = 400 \)
- When \( t = 7 \): \( A(7) = 450 \)
- For \( B(t) \):
- When \( t = 0 \): \( B(0) = 200 \)
- When \( t = 1 \): \( B(1) = 225 \)
- When \( t = 2 \): \( B(2) = 250 \)
- When \( t = 3 \): \( B(3) = 275 \)
- When \( t = 4 \): \( B(4) = 300 \)
- When \( t = 5 \): \( B(5) = 325 \)
- When \( t = 6 \): \( B(6) = 350 \)
- When \( t = 7 \): \( B(7) = 375 \)
3. **Plot the points on a graph:**
- Plot the points calculated for both functions on a graph. Use different colors or markers for each company.
- For Company A, the points would be plotted like (0, 100), (1, 150), (2, 200), (3, 250), (4, 300), (5, 350), (6, 400), (7, 450).
- For Company B, the points would be plotted like (0, 200), (1, 225), (2, 250), (3, 275), (4, 300), (5, 325), (6, 350), (7, 375).
4. **Connect the points with lines:**
- Draw a straight line through the points for Company A which will have a steeper slope due to the higher hourly rate.
- Draw a straight line through the points for Company B which will have a gentler slope since their hourly rate is lower.
5. **Label the graph:**
- Label the x-axis (Hours of Landscaping) and y-axis (Total Cost).
- Include a legend to differentiate between Company A and Company B.
- Mark the intersection point (4, 300) where both companies charge the same price.
### Visual Representation:
The graph would reflect:
- Company A's line starts at (0, 100) and rises steeply, crossing through points like (4, 300) and (7, 450).
- Company B's line starts higher at (0, 200) but rises more gently, crossing the same point (4, 300) and reaching (7, 375).
### Key Points to Observe:
- The intersection point at \( t = 4 \) hours is significant, indicating the switch in cost-effectiveness between the two companies.
- For times less than 4 hours, Company B is cheaper (e.g., at 3 hours, Company A is $250, and Company B is $275).
- For times greater than 4 hours, Company A becomes cheaper (e.g., at 5 hours, Company A is $350, and Company B is $325).
By following these steps, you can effectively create a graphical representation of the cost comparison between the two landscaping companies for a range of hours worked.
### Step-by-step Instructions to Graph \( A(t) \) and \( B(t) \):
1. **Identify the functions:**
- Company A: \( A(t) = 50t + 100 \)
- Company B: \( B(t) = 25t + 200 \)
2. **Determine key points:**
- For \( A(t) \):
- When \( t = 0 \): \( A(0) = 100 \)
- When \( t = 1 \): \( A(1) = 150 \)
- When \( t = 2 \): \( A(2) = 200 \)
- When \( t = 3 \): \( A(3) = 250 \)
- When \( t = 4 \): \( A(4) = 300 \)
- When \( t = 5 \): \( A(5) = 350 \)
- When \( t = 6 \): \( A(6) = 400 \)
- When \( t = 7 \): \( A(7) = 450 \)
- For \( B(t) \):
- When \( t = 0 \): \( B(0) = 200 \)
- When \( t = 1 \): \( B(1) = 225 \)
- When \( t = 2 \): \( B(2) = 250 \)
- When \( t = 3 \): \( B(3) = 275 \)
- When \( t = 4 \): \( B(4) = 300 \)
- When \( t = 5 \): \( B(5) = 325 \)
- When \( t = 6 \): \( B(6) = 350 \)
- When \( t = 7 \): \( B(7) = 375 \)
3. **Plot the points on a graph:**
- Plot the points calculated for both functions on a graph. Use different colors or markers for each company.
- For Company A, the points would be plotted like (0, 100), (1, 150), (2, 200), (3, 250), (4, 300), (5, 350), (6, 400), (7, 450).
- For Company B, the points would be plotted like (0, 200), (1, 225), (2, 250), (3, 275), (4, 300), (5, 325), (6, 350), (7, 375).
4. **Connect the points with lines:**
- Draw a straight line through the points for Company A which will have a steeper slope due to the higher hourly rate.
- Draw a straight line through the points for Company B which will have a gentler slope since their hourly rate is lower.
5. **Label the graph:**
- Label the x-axis (Hours of Landscaping) and y-axis (Total Cost).
- Include a legend to differentiate between Company A and Company B.
- Mark the intersection point (4, 300) where both companies charge the same price.
### Visual Representation:
The graph would reflect:
- Company A's line starts at (0, 100) and rises steeply, crossing through points like (4, 300) and (7, 450).
- Company B's line starts higher at (0, 200) but rises more gently, crossing the same point (4, 300) and reaching (7, 375).
### Key Points to Observe:
- The intersection point at \( t = 4 \) hours is significant, indicating the switch in cost-effectiveness between the two companies.
- For times less than 4 hours, Company B is cheaper (e.g., at 3 hours, Company A is $250, and Company B is $275).
- For times greater than 4 hours, Company A becomes cheaper (e.g., at 5 hours, Company A is $350, and Company B is $325).
By following these steps, you can effectively create a graphical representation of the cost comparison between the two landscaping companies for a range of hours worked.