The Hourly Parking Garage at BWI International Airport in Baltimore costs $2 per half hour during the first hour and $4 for each hour or part of an hour for more than 1 hour and up to 5 hours. If
C = C(t)
is the total charge for t hours in BWI's Hourly Parking Garage, write a piecewise function that describes
C(t)
for
t < 3.
Use
C(t)
to find the following limits, if they exist. (If an answer does not exist, enter DNE.)
(a)
lim
t→0.5−
C(t)

$

(b)
lim
t→0.5+
C(t)

$

(c)
lim
t→0.5
C(t)

$

(d)
lim
t→2.5
C(t)

$

We are tasked with defining a piecewise function $ C(t) $ that models the total parking charge at BWI International Airport’s Hourly Parking Garage for $ t < 3 $, where $ t $ represents the number of hours parked. Based on the pricing structure:

- **$2 per half-hour** during the first hour (i.e., $ 0 \leq t \leq 1 $).
- **$4 for each hour or part of an hour** for more than 1 hour up to 5 hours (i.e., $ 1 < t \leq 5 $).

We need to find limits as $ t $ approaches certain values, and to do that, we will define the piecewise function.

### Step 1: Define the piecewise function $ C(t) $

For $ t \leq 1 $, the charge is based on the half-hour increments:
\[
C(t) =
\begin{cases} 
2t, & 0 \leq t \leq 1 \\
2 + 4(t - 1), & 1 < t \leq 3 
\end{cases}
\]
- When $ 0 \leq t \leq 1 $, the cost is $ 2t $ dollars (since it's $2 per half-hour, so $4 per hour).
- When $ 1 < t \leq 3 $, the cost is $2 for the first hour plus $4 per additional hour (or part of an hour). This simplifies to $ 2 + 4(t - 1) $.

### Step 2: Evaluate the limits

Now, let’s evaluate the limits requested in the problem:

#### (a) $ \lim_{t \to 0.5^-} C(t) $

As $ t \to 0.5 $ from the left ($ t \to 0.5^- $), we use the first part of the piecewise function $ C(t) = 2t $ for $ 0 \leq t \leq 1 $:
\[
\lim_{t \to 0.5^-} C(t) = 2(0.5) = 1.
\]

Thus, the limit is:
\[
\boxed{1}.
\]

#### (b) $ \lim_{t \to 0.5^+} C(t) $

As $ t \to 0.5 $ from the right ($ t \to 0.5^+ $), we are still in the same interval $ 0 \leq t \leq 1 $, where $ C(t) = 2t $:
\[
\lim_{t \to 0.5^+} C(t) = 2(0.5) = 1.
\]

Thus, the limit is:
\[
\boxed{1}.
\]

#### (c) $ \lim_{t \to 0.5} C(t) $

Since both the left-hand limit and the right-hand limit are equal, the limit at $ t = 0.5 $ exists and is:
\[
\boxed{1}.
\]

#### (d) $ \lim_{t \to 2.5} C(t) $

As $ t \to 2.5 $, we use the second part of the piecewise function $ C(t) = 2 + 4(t - 1) $ for $ 1 < t \leq 3 $. Substituting $ t = 2.5 $:
\[
C(2.5) = 2 + 4(2.5 - 1) = 2 + 4(1.5) = 2 + 6 = 8.
\]

Thus, the limit is:
\[
\boxed{8}.
\]

### Summary of the Limits:
- (a) $ \lim_{t \to 0.5^-} C(t) = 1 $
- (b) $ \lim_{t \to 0.5^+} C(t) = 1 $
- (c) $ \lim_{t \to 0.5} C(t) = 1 $
- (d) $ \lim_{t \to 2.5} C(t) = 8 $

Would you like to go over any of these steps in more detail?

---

Here are five related questions you could explore:
1. What happens to the limit as $ t \to 1^- $ and $ t \to 1^+ $?
2. How would the piecewise function change if the pricing structure allowed for $3 per half-hour instead of $2?
3. What is $ \lim_{t \to 3} C(t) $ based on the current function definition?
4. How can you use derivatives to find the rate of change of the parking cost at different times?
5. How does the limit change if the maximum parking time is reduced to 2 hours?

**Tip:** When working with piecewise functions, always check which part of the function applies for different limit directions.

The Hourly Parking Garage at BWI International Airport in Baltimore costs $2 per half hour during the first hour and $4 for each hour or part of an hour for more than 1 hour and up to 5 hours. If C = C(t) is the total charge for t hours in BWI's Hourly Parking Garage, write a piecewise function that describes C(t) for t < 3. Use C(t) to find the following limits, if they exist. (If an answer does not exist, enter DNE.) (a) lim t→0.5− C(t) (b) lim t→0.5+ C(t) (c) lim t→0.5 C(t) (d) lim t→2.5 C(t)

Explore how to evaluate limits of a piecewise function that models parking charges at BWI International Airport. This problem involves finding the limits of C(t) as t approaches specific values and covers core concepts in calculus. Suitable for students in Grades 11-12.

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