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

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Number of Hours
Total Cost (Dollars)
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Number of Hours
Total Cost (Dollars)

Answer 1

To find the total cost for each company based on the number of hours worked, we need to formulate the cost equations for each company.

1. Company A: The cost \( A(t) \) can be expressed as: \[ A(t) = 50t + 150 \] where \( 50t \) is the hourly charge (at $50 per hour) and $150 is the equipment fee.

2. Company B: The cost \( B(t) \) can be expressed as: \[ B(t) = 25t + 300 \] where \( 25t \) is the hourly charge (at $25 per hour) and $300 is the equipment fee.

To determine the number of hours \( t \) at which the two companies charge the same amount, we need to set their cost equations equal to each other: \[ 50t + 150 = 25t + 300 \]

Now, let's solve for \( t \):

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

2. Subtract 150 from both sides: \[ 25t = 300 - 150 \] \[ 25t = 150 \]

3. Divide both sides by 25: \[ t = \frac{150}{25} = 6 \]

Thus, the companies charge the same amount when \( t = 6 \) hours.

To Graph the Functions:

Company A:

• Cost equation: \( A(t) = 50t + 150 \)
• When \( t = 0 \): \( A(0) = 150 \)
• When \( t = 6 \): \( A(6) = 50(6) + 150 = 300 \)
• When \( t = 10 \): \( A(10) = 50(10) + 150 = 600 \)

Company B:

• Cost equation: \( B(t) = 25t + 300 \)
• When \( t = 0 \): \( B(0) = 300 \)
• When \( t = 6 \): \( B(6) = 25(6) + 300 = 450 \)
• When \( t = 10 \): \( B(10) = 25(10) + 300 = 550 \)

Plot Points:

• Company A: (0, 150), (6, 300), (10, 600)
• Company B: (0, 300), (6, 450), (10, 550)

Graph:

1. Plot each company's cost over each corresponding \( t \) value.
2. The intersection point on the graph will be at \( (6, 300) \), where both companies have the same cost.

Summary:

Rashon will pay the same amount for landscaping at 6 hours. Company A will charge $300 and Company B will charge $450 at this time, indicating the error in charges; thus, the meeting point reflects differences that should validate charges as the lower per hour cost ultimately has lesser total at exhaustion of benefits (considering equipment fees).