To model the cost of renting roller skates based on the given time intervals, we need to define a piecewise function [tex]\( c(t) \)[/tex] where [tex]\( t \)[/tex] is the rental time in hours, and [tex]\( c(t) \)[/tex] is the cost in dollars. We'll break down the intervals step by step as per the given cost structure:
1. If [tex]\( 0 < t \leq 1 \)[/tex]:
For any time [tex]\( t \)[/tex] between 0 and 1 hour, the cost [tex]\( c(t) \)[/tex] is [tex]\( \$5 \)[/tex].
2. If [tex]\( 1 < t \leq 2 \)[/tex]:
For any time [tex]\( t \)[/tex] between more than 1 hour and up to 2 hours, the cost [tex]\( c(t) \)[/tex] increases to [tex]\( \$10 \)[/tex].
3. If [tex]\( 2 < t \leq 5 \)[/tex]:
For any time [tex]\( t \)[/tex] between more than 2 hours and up to 5 hours, the cost [tex]\( c(t) \)[/tex] further increases to [tex]\( \$20 \)[/tex].
4. If [tex]\( 5 < t \leq 8 \)[/tex]:
For any time [tex]\( t \)[/tex] between more than 5 hours and up to 8 hours, the cost [tex]\( c(t) \)[/tex] is [tex]\( \$25 \)[/tex].
For [tex]\( t \)[/tex] values beyond 8 hours, we assume the rental time exceeds the defined limits, so the cost function does not cover these values.
Given this structure, the piecewise function [tex]\( c(t) \)[/tex] can be written as:
[tex]\[
c(t) = \begin{cases}
5 & \text{if } 0 < t \leq 1 \\
10 & \text{if } 1 < t \leq 2 \\
20 & \text{if } 2 < t \leq 5 \\
25 & \text{if } 5 < t \leq 8 \\
\text{Time exceeds limits} & \text{if } t > 8
\end{cases}
\][/tex]
This function reflects the cost structure provided by the skate shop. Each interval corresponds to a specific rental cost, ensuring the model accurately represents the rental pricing for up to 8 hours within a single day. The function also indicates that any rental time beyond 8 hours would exceed the stipulated rental periods provided.
