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Cara likes candles. She also likes mathematics and was thinking about using algebra to answer a question she had about two of her candles. Her taller candle is 16 centimeters tall. Each hour it burns, the candle loses 2.5 centimeters in height. Her shorter candle is 12 centimeters tall and loses 1.5 centimeters in height for each hour that it burns.

Cara started filling out the following table to help determine whether these two candles would ever reach the same height at the same time if allowed to burn for the same length of time. Finish the table for Cara.

\begin{tabular}{|c|c|c|}
\hline
Time (hours) & 16 cm candle height (cm) & 12 cm candle height (cm) \\
\hline
0 & 16 & 12 \\
\hline
1 & 13.5 & 10.5 \\
\hline
2 & & \\
\hline
3 & & \\
\hline
4 & & \\
\hline
5 & & \\
\hline
6 & & \\
\hline
7 & & \\
\hline
& & \\
\hline
& & \\
\hline
\end{tabular}

Use the data in the table to identify a function in slope-intercept form for the height, [tex]$S$[/tex], of the smaller candle in terms of the number of hours it has burned, [tex]$h$[/tex].

A. [tex]$S=12-1.5h$[/tex]
B. [tex]$S=-12+1.5h$[/tex]
C. [tex]$S=12+1.5h$[/tex]
D. [tex]$S=-12-1.5h$[/tex]



Answer :

GinnyAnswer
To answer the question in detail, let us:

1. Analyze the given data points for the candles.
2. Use this data to determine the function in slope-intercept form for the height of the smaller candle over time.

### Analyzing the Given Data Points

- Initial Height and Burning Rate:
- Taller Candle: Initial height = 16 cm, burns at a rate of 2.5 cm/hour.
- Smaller Candle: Initial height = 12 cm, burns at a rate of 1.5 cm/hour.

### Filling out the Table

Calculations for the Taller Candle:
- Height \( H_t \) at time \( h \):
\( H_t = 16 - 2.5h \)

Calculations for the Smaller Candle:
- Height \( H_s \) at time \( h \):
\( H_s = 12 - 1.5h \)

Using these equations, we can fill out the table:

[tex]\[ \begin{tabular}{|c|c|c|} \hline \textbf{Time (hours)} & \textbf{Taller Candle Height (cm)} & \textbf{Smaller Candle Height (cm)} \\ \hline 0 & 16 & 12 \\ \hline 1 & 13.5 & 10.5 \\ \hline 2 & 11 & 9 \\ \hline 3 & 8.5 & 7.5 \\ \hline 4 & 6 & 6 \\ \hline 5 & 3.5 & 4.5 \\ \hline 6 & 1 & 3 \\ \hline 7 & -1.5 & 1.5 \\ \hline \end{tabular} \][/tex]

### Identifying the Function for the Smaller Candle

The height of the smaller candle can be described by the linear equation \( S = 12 - 1.5h \), where \( S \) represents the height in centimeters and \( h \) represents the time in hours.

Given the provided options:
1. \( S = 12 - 1.5h \) (Correct)
2. \( S = -12 + 1.5h \)
3. \( S = 12 + 1.5h \)
4. \( S = -12 - 1.5h \)

The correct slope-intercept form equation for the height of the smaller candle is:
[tex]\[ \boxed{S = 12 - 1.5h} \][/tex]

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