Questions: Suppose you have a turtle racing. It starts at the start line and goes 2 feet per second for the initial 3 seconds. After that it speeds up to 3 feet per second. (a) Create a table for this turtle's distance from the starting line after t seconds. Assume the race ends 12 feet from the starting line. t d(t) ------ 0 0 1 2 3 4 5 (b) What is the equation and constraint for the first part of this turtle's race?

Solution Steps

To solve this problem, we need to calculate the distance the turtle travels at each second, given its speed changes after 3 seconds. For part (a), we will create a table showing the distance at each second up to 5 seconds. For part (b), we will determine the equation for the distance during the first part of the race and specify the constraint.

1. For part (a), calculate the distance for each second using the given speeds: 2 feet per second for the first 3 seconds, and 3 feet per second thereafter.
2. For part (b), the equation for the first part of the race is a linear equation based on the speed of 2 feet per second, with the constraint that time \( t \) is between 0 and 3 seconds.

We calculated the distance \( d(t) \) of the turtle from the starting line at each second \( t \) from 0 to 5 seconds. The distances are as follows:

\[ \begin{array}{|c|c|} \hline t & d(t) \\ \hline 0 & 0 \\ 1 & 2 \\ 2 & 4 \\ 3 & 6 \\ 4 & 9 \\ 5 & 12 \\ \hline \end{array} \]

For the first part of the turtle's race, where it travels at a speed of 2 feet per second, the equation for the distance is given by:

\[ d(t) = 2t \quad \text{for } 0 \leq t \leq 3 \]

The constraint for this part of the race is:

\[ 0 \leq t \leq 3 \]

Final Answer