Question

Mason is deciding between two parking garages. Garage A charges an initial fee of $12 to park plus $5 per hour. Garage B charges an initial fee of $3 to park plus $6 per hour. Let AA represent the amount Garage A would charge if Mason parks for tt hours, and let BB represent the amount Garage B would charge if Mason parks for tt hours. Write an equation for each situation, in terms of t, commat, and determine the hours parked, t, commat, that would make the cost of each garage the same.

Answer 1

To model the cost of parking in each garage, we can define the equations for each garage based on the information provided:

**Garage A:**
The total cost \( A \) for parking at Garage A can be expressed as:
\[
A = 12 + 5t
\]
Here, \( 12 \) is the initial fee, and \( 5t \) is the hourly charge based on the number of hours \( t \).

**Garage B:**
The total cost \( B \) for parking at Garage B can be expressed as:
\[
B = 3 + 6t
\]
Here, \( 3 \) is the initial fee, and \( 6t \) is the hourly charge based on the number of hours \( t \).

To find the number of hours \( t \) at which both garages charge the same amount, we set the two equations equal to each other:
\[
12 + 5t = 3 + 6t
\]

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

1. Subtract \( 5t \) from both sides:
\[
12 = 3 + 6t - 5t
\]
Which simplifies to:
\[
12 = 3 + t
\]

2. Subtract \( 3 \) from both sides:
\[
12 - 3 = t
\]
Therefore:
\[
t = 9
\]

So, the number of hours \( t \) that would make the cost of each garage the same is \( \boxed{9} \) hours.