Question

Bo is deciding between two parking garages. Garage A charges an initial fee of $4 to park plus $2.50 per hour. Garage B charges an initial fee of $9 to park plus $1.50 per hour. Let AA represent the amount Garage A would charge if Bo parks for tt hours, and let BB represent the amount Garage B would charge if Bo parks for tt hours. Graph each function and determine which garage would be cheaper if Bo needs to park for 4 hours.

Answer 1

To analyze the parking costs from both garages, we can define the cost functions for Garage A and Garage B.

1. **Garage A**:
- Initial fee: $4
- Hourly charge: $2.50 per hour
- The cost function for Garage A can be expressed as:
\[
A(t) = 4 + 2.5t
\]

2. **Garage B**:
- Initial fee: $9
- Hourly charge: $1.50 per hour
- The cost function for Garage B can be expressed as:
\[
B(t) = 9 + 1.5t
\]

Next, we will calculate the costs for each garage if Bo parks for \( t = 4 \) hours.

### Calculating the Costs
- For Garage A:
\[
A(4) = 4 + 2.5 \cdot 4 = 4 + 10 = 14
\]

- For Garage B:
\[
B(4) = 9 + 1.5 \cdot 4 = 9 + 6 = 15
\]

### Summary of Costs
- Cost at Garage A for 4 hours: $14
- Cost at Garage B for 4 hours: $15

### Conclusion
Garage A would be cheaper if Bo needs to park for 4 hours, as it will cost him $14, compared to the $15 he would pay at Garage B.

### Graphing the Functions
To graph the functions, you would plot the lines defined by the equations over a reasonable time interval. For example, you could plot \( t \) (the number of hours parked) on the x-axis and the respective costs \( A(t) \) and \( B(t) \) on the y-axis.

- **Garage A** would be a line starting at (0, 4) and have a slope of 2.5.
- **Garage B** would start at (0, 9) and have a slope of 1.5.

The intersection points and points at \( t = 4 \) will help confirm the comparative costs visually.