To find the cost for each company, we can express the total cost as a function of the number of hours \( t \).
Company A:
The total cost for Company A can be formulated as: \[ A(t) = 50t + 150 \]
Company B:
The total cost for Company B can be formulated as: \[ B(t) = 25t + 300 \]
Next, we need to determine the point where the costs are equal, which means solving the equation \( A(t) = B(t) \). Setting the two equations equal gives us:
\[ 50t + 150 = 25t + 300 \]
Now, we will solve for \( t \):
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Subtract \( 25t \) from both sides: \[ 50t - 25t + 150 = 300 \] This simplifies to: \[ 25t + 150 = 300 \]
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Subtract 150 from both sides: \[ 25t = 300 - 150 \] This simplifies to: \[ 25t = 150 \]
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Divide both sides by 25: \[ t = \frac{150}{25} = 6 \]
Thus, the number of hours \( t \) that would make the cost of each company the same is \( t = 6 \) hours.
Summary of Costs
Now let's calculate the total costs for both companies:
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For Company A at \( t = 6 \): \[ A(6) = 50(6) + 150 = 300 + 150 = 450 \]
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For Company B at \( t = 6 \): \[ B(6) = 25(6) + 300 = 150 + 300 = 450 \]
Both companies would charge $450 for 6 hours of landscaping.
Graphing the Functions
To graph these functions:
- X-axis: Number of hours \( t \).
- Y-axis: Total cost \( A(t) \) and \( B(t) \).
The lines will have the following characteristics:
- The line for Company A has a steeper slope (50) and starts at \( 150 \) on the Y-axis.
- The line for Company B has a gentler slope (25) but starts higher at \( 300 \) on the Y-axis.
You can graph these equations to visualize where they intersect — the intersection point is at \( (6, 450) \). This point represents the number of hours where the costs for both landscaping companies are the same.
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