Question

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 Addison 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 parking garage charges, we can set up equations based on the pricing structures provided.

For Garage A:

• Initial fee = $12
• Fee per hour = $5

Thus, the total cost for Garage A, \( A(t) \), for parking \( t \) hours can be expressed as: \[ A(t) = 12 + 5t \]

For Garage B:

• Initial fee = $7
• Fee per hour = $6

Thus, the total cost for Garage B, \( B(t) \), for parking \( t \) hours can be expressed as: \[ B(t) = 7 + 6t \]

Now, we want to find the interval of hours \( t \) for which Garage A is cheaper than Garage B. This means we need to solve the inequality: \[ A(t) < B(t) \]

Substituting the expressions for \( A(t) \) and \( B(t) \): \[ 12 + 5t < 7 + 6t \]

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

1. Start by isolating the terms with \( t \) on one side: \[ 12 < 7 + 6t - 5t \]
2. Simplify: \[ 12 < 7 + t \]
3. Subtract 7 from both sides: \[ 12 - 7 < t \] \[ 5 < t \] Thus, we can rewrite this as: \[ t > 5 \]

This result tells us that Garage A is cheaper than Garage B when \( t \) (the number of hours parked) is greater than 5.

In conclusion:

• The inequalities for the costs are \( A(t) = 12 + 5t \) and \( B(t) = 7 + 6t \).
• Garage A is cheaper than Garage B when \( t > 5 \) hours.