Addy 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 $7 to park plus $6 per hour. Let AA represent the amount Garage A would charge if Addy parks for tt hours, and let BB represent the amount Garage B would charge if Addy parks for tt hours. Write an equation for each situation, in terms of t, commat, and determine the interval of hours parked, t, commat, for which Garage A is cheaper than Garage B.

Answer 1

To find out how much each garage charges based on the number of hours parked, we can write equations for the costs of Garage A and Garage B in terms of \( t \), where \( t \) is the number of hours parked.

Garage A:

• Initial fee: $12
• Rate per hour: $5

The equation for the cost of parking in Garage A is: \[ A(t) = 12 + 5t \]

Garage B:

• Initial fee: $7
• Rate per hour: $6

The equation for the cost of parking in Garage B is: \[ B(t) = 7 + 6t \]

Next, we want to determine the interval of hours \( t \) for which Garage A is cheaper than Garage B. This can be set up as an inequality: \[ A(t) < B(t) \]

Substituting the equations we have: \[ 12 + 5t < 7 + 6t \]

Now, we can solve for \( t \):

1. Subtract \( 5t \) from both sides: \[ 12 < 7 + t \]

2. Subtract 7 from both sides: \[ 5 < t \]

Thus, we can express this as: \[ t > 5 \]

Therefore, Garage A is cheaper than Garage B when Addy parks for more than 5 hours.

In conclusion, the interval of hours parked \( t \) for which Garage A is cheaper than Garage B is: \[ (5, \infty) \]