Question

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.

Answer 1

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

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

3. 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:

- For Company A at \( t = 6 \):
\[
A(6) = 50(6) + 150 = 300 + 150 = 450
\]

- 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:

1. **X-axis**: Number of hours \( t \).
2. **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.