College

You began membership in a new health and fitness club, which gives you access to a dietician and a personal trainer. They help you develop a special eight-week diet and exercise program. The data in the following table represents your weight, [tex]w[/tex], as a function of time, [tex]t[/tex], over an eight-week period.

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

Using [tex]\Delta[/tex] notation, determine the average rate of change of weight over the last four weeks of the program (include weeks 4 and 8 in your calculations).

A. [tex]-2.75[/tex] pounds per week
B. [tex]+2.75[/tex] pounds per week
C. [tex]-4.0[/tex] pounds per week
D. [tex]+0.38[/tex] pounds per week

Please select the best answer from the choices provided.

Answer :

To find the average rate of change of weight over the last four weeks of the program (from week 4 to week 8), we'll follow these steps:

1. Identify the data points we need:
- At week 4, the weight is 140 pounds.
- At week 8, the weight is 129 pounds.

2. Calculate the change in weight ([tex]\(\Delta \text{weight}\)[/tex]):
- Subtract the weight at week 4 from the weight at week 8.
[tex]\[
\Delta \text{weight} = \text{Weight at week 8} - \text{Weight at week 4} = 129 - 140 = -11 \text{ pounds}
\][/tex]

3. Calculate the change in time ([tex]\(\Delta \text{time}\)[/tex]):
- Subtract week 4 from week 8.
[tex]\[
\Delta \text{time} = 8 - 4 = 4 \text{ weeks}
\][/tex]

4. Calculate the average rate of change:
- Divide the change in weight by the change in time.
[tex]\[
\text{Average rate of change} = \frac{\Delta \text{weight}}{\Delta \text{time}} = \frac{-11 \text{ pounds}}{4 \text{ weeks}} = -2.75 \text{ pounds per week}
\][/tex]

Therefore, the average rate of change of weight over the last four weeks of the program is [tex]\(-2.75\)[/tex] pounds per week.

The correct answer from the choices provided is option a: [tex]\(-2.75\)[/tex] pounds per week.