Select the correct answer.

A chemist is performing an experiment by observing the effects when she combines two solutions. She monitors the temperature of the combined solution over the course of eight hours and records it in this table.

\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|}
\hline Time (hours) & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline Temperature [tex]$\left({ }^{\circ} F \right)$[/tex] & 20.50 & 19.82 & 17.42 & 14.02 & 10.82 & 9.50 & 12.22 & 21.62 & 40.82 \\
\hline
\end{tabular}

The chemist can model the temperature with a polynomial function. During these eight hours, over what interval is the temperature decreasing?

A. from hour 0 to hour 8
B. from hour 2 to hour 6
C. from hour 5 to hour 8
D. from hour 0 to hour 5



Answer :

To determine the interval during which the temperature is decreasing, we need to analyze the temperature changes between consecutive hours based on the provided data:

[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline \text{Time (hours)} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \text{Temperature (}^{\circ}\text{F)} & 20.50 & 19.82 & 17.42 & 14.02 & 10.82 & 9.50 & 12.22 & 21.62 & 40.82 \\ \hline \end{array} \][/tex]

We will calculate the difference in temperature between each consecutive hour:

1. Difference from hour 0 to hour 1: [tex]\(19.82 - 20.50 = -0.68\)[/tex]
2. Difference from hour 1 to hour 2: [tex]\(17.42 - 19.82 = -2.40\)[/tex]
3. Difference from hour 2 to hour 3: [tex]\(14.02 - 17.42 = -3.40\)[/tex]
4. Difference from hour 3 to hour 4: [tex]\(10.82 - 14.02 = -3.20\)[/tex]
5. Difference from hour 4 to hour 5: [tex]\(9.50 - 10.82 = -1.32\)[/tex]
6. Difference from hour 5 to hour 6: [tex]\(12.22 - 9.50 = +2.72\)[/tex]
7. Difference from hour 6 to hour 7: [tex]\(21.62 - 12.22 = +9.40\)[/tex]
8. Difference from hour 7 to hour 8: [tex]\(40.82 - 21.62 = +19.20\)[/tex]

We observe the following temperature trends:

- Decreasing from hour 0 to hour 5
- Increasing from hour 5 to hour 6
- Increasing from hour 6 to hour 7
- Increasing from hour 7 to hour 8

Thus, the interval where the temperature consistently decreases is from hour 0 to hour 5.

Therefore, the correct answer is:
D. from hour 0 to hour 5