Rashon is deciding between two landscaping companies for his place of business. Company A charges $50 per hour and a $150 equipment fee. Company B charges $25 per hour and a $300 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 the number hours, t, commat, that would make the cost of each company the same.

Question

make sure to include how to mark the graph and what to mark on it

Answer 1

To analyze the costs of the two landscaping companies, we can represent each company's pricing structure as a linear function based on the number of hours worked, \( t \).

Cost Functions

1. Company A: \[ A(t) = 50t + 150 \]

   • Here, \( A(t) \) signifies the total cost from Company A after \( t \) hours of work. The slope is 50 (the hourly rate), and the y-intercept is 150 (the equipment fee).

2. Company B: \[ B(t) = 25t + 300 \]

   • Similarly, \( B(t) \) represents the total cost from Company B after \( t \) hours of work, where the slope is 25 and the y-intercept is 300.

Finding the Intersection Point

To determine the number of hours \( t \) when the costs are equal (i.e., \( A(t) = B(t) \)), we set the two equations equal to each other:

\[ 50t + 150 = 25t + 300 \]

Solving for \( t \):

1. Subtract \( 25t \) from both sides: \[ 50t - 25t + 150 = 300 \] \[ 25t + 150 = 300 \]

2. Subtract 150 from both sides: \[ 25t = 150 \]

3. Divide by 25: \[ t = 6 \]

Graphing the Functions

To graph the functions \( A(t) \) and \( B(t) \):

1. Choose a range for \( t \): For this context, let's plot up to \( t = 10 \) hours.

2. Calculate values:

   • For \( t = 0 \) to \( t = 10 \):

     • For Company A:

       • \( A(0) = 150 \)
       • \( A(1) = 200 \)
       • \( A(2) = 250 \)
       • \( A(3) = 300 \)
       • \( A(4) = 350 \)
       • \( A(5) = 400 \)
       • \( A(6) = 450 \)
       • \( A(7) = 500 \)
       • \( A(8) = 550 \)
       • \( A(9) = 600 \)
       • \( A(10) = 650 \)

     • For Company B:

       • \( B(0) = 300 \)
       • \( B(1) = 325 \)
       • \( B(2) = 350 \)
       • \( B(3) = 375 \)
       • \( B(4) = 400 \)
       • \( B(5) = 425 \)
       • \( B(6) = 450 \)
       • \( B(7) = 475 \)
       • \( B(8) = 500 \)
       • \( B(9) = 525 \)
       • \( B(10) = 550 \)

3. Plot the points:

   • On a Cartesian plane, mark the x-axis for \( t \) (hours) and the y-axis for cost (dollars).
   • Plot the values calculated above for both Company A and Company B.
   • Use different colors or markers for Company A and B for clarity.

Marking the Graph

• Label the axes:

  • X-axis: "Hours of Landscaping (t)"
  • Y-axis: "Cost (in dollars)"

• Title the graph: "Cost Comparison of Landscaping Companies A and B"

• Highlight the point where the two lines intersect at \( (6, 450) \). This denotes that at 6 hours, both companies will charge $450.

Conclusion

From the analysis and graphing of the two functions, we conclude that Rashon would have the same cost for both Company A and Company B when he hires them for 6 hours of landscaping services.