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.

Question

GPT-4o mini · Answer

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.

For Company A: \[ A(7) = 50 \times 7 + 100 = 350 + 100 = 450 \]
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.

Answer

how would that be marked on a graph

GPT-4o mini · Answer

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) $:

Identify the functions:
- Company A: $ A(t) = 50t + 100 $
- Company B: $ B(t) = 25t + 200 $
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 $
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).
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.
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.