The weight, [tex] w [/tex], as a function of time, [tex] t [/tex], over an eight-week period is given below:

[tex]
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\text{Time (weeks)} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
\text{Weight (lb)} & 150 & 144 & 143 & 141 & 140 & 137 & 137 & 132 & 129 \\
\hline
\end{array}
[/tex]

Write the ratio of the change in weight for the first five weeks to the change in time for the first five weeks and interpret the meaning of this ratio.

A. [tex]\frac{-13}{5}[/tex], 13 pounds were lost in 5 weeks
B. [tex]\frac{5}{13}[/tex], 13 pounds were gained in 5 weeks
C. [tex]\frac{-13}{5}[/tex], 13 pounds were lost in 5 weeks
D. [tex]\frac{5}{-13}[/tex], 5 pounds were lost in 13 weeks

Let's break down the problem to understand the change in weight over the first five weeks and interpret its meaning.

1. Initial Information:
- We have a list of weights recorded weekly.
- We need to examine the change in weight during the first five weeks.

2. Extract Relevant Data:
- Initial weight (Week 0): 150 pounds
- Weight after five weeks (Week 5): 137 pounds

3. Calculate Change in Weight:
- The change in weight over these five weeks is:
[tex]\[
\text{Change in Weight} = \text{Weight after five weeks} - \text{Initial weight} = 137 - 150 = -13 \text{ pounds}
\][/tex]

4. Calculate Change in Time:
- The change in time over the first five weeks is from week 0 to week 5:
[tex]\[
\text{Change in Time} = 5 - 0 = 5 \text{ weeks}
\][/tex]

5. Form the Ratio:
- We compare the change in weight to the change in time:
[tex]\[
\text{Ratio} = \frac{\text{Change in Weight}}{\text{Change in Time}} = \frac{-13}{5}
\][/tex]

6. Interpret the Ratio:
- This ratio, [tex]\(\frac{-13}{5}\)[/tex], means that a total of 13 pounds was lost over 5 weeks. Negative sign indicates weight loss.

The correct interpretation and ratio match option C:
- [tex]\(\frac{-13}{5}, 13\)[/tex] pounds was lost in 5 weeks.