Question 1164606: Write a piecewise function for the given situation.
A parking garage charges $2 an hour for parking in the garage with a maximum amount of money you can be charged is $16 no matter the amount time you parked there.
Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website! To write this piecewise function, we define the cost $C$ based on the number of hours $t$ spent in the garage.
### **1. Identify the Conditions**
* **The Hourly Rate:** The garage charges **$2$** per hour. This is represented by $2t$.
* **The Maximum Cap:** Once the cost reaches **$16$**, the price stops increasing.
* **The "Switch" Point:** To find when the cap kicks in, we solve $2t = 16$. This happens at **$t = 8$** hours.
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### **2. The Piecewise Function**
Let $C(t)$ be the total cost in dollars and $t$ be the time in hours ($t \ge 0$):
$$C(t) =
\begin{cases}
2t & 0 \le t < 8 \\
16 & t \ge 8
\end{cases}$$
---
### **3. Breakdown of the Function**
* **$2t$ for $0 \le t < 8$:** If you stay for less than 8 hours, you pay exactly \$2 for every hour parked. For example, 3 hours costs \$6.
* **$16$ for $t \ge 8$:** Once you hit the 8-hour mark, the "flat rate" applies. Whether you stay for 9 hours, 12 hours, or 24 hours, the price remains stuck at \$16.
### **Quick Check**
If you stay for **5 hours**: $C(5) = 2(5) = \$10$.
If you stay for **10 hours**: $C(10) = \$16$ (since $10 > 8$).
Does your specific parking problem require you to account for partial hours (like rounding up to the next hour), or is it strictly linear?
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