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Cara likes candles. She also likes mathematics and was thinking about using algebra to answer a question that 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 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) & [tex]$16 \, \text{cm}$[/tex] candle height [tex]$(\text{cm})$[/tex] & [tex]$12 \, \text{cm}$[/tex] candle height [tex]$(\text{cm})$[/tex] \\
\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}



Answer :

GinnyAnswer
Let's fill out the rest of the table step-by-step.

We start by noting the initial heights and the burn rates of the two candles:

1. The taller candle starts at 16 cm and loses 2.5 cm in height each hour.
2. The shorter candle starts at 12 cm and loses 1.5 cm in height each hour.

Next, let's calculate the heights for each candle from hour 2 to hour 7.

### Step-by-Step Calculation

- Time (hours) = 2:
- Tall candle height: \( 16 - 2 \times 2.5 = 16 - 5 = 11 \) cm
- Short candle height: \( 12 - 2 \times 1.5 = 12 - 3 = 9 \) cm

- Time (hours) = 3:
- Tall candle height: \( 16 - 3 \times 2.5 = 16 - 7.5 = 8.5 \) cm
- Short candle height: \( 12 - 3 \times 1.5 = 12 - 4.5 = 7.5 \) cm

- Time (hours) = 4:
- Tall candle height: \( 16 - 4 \times 2.5 = 16 - 10 = 6 \) cm
- Short candle height: \( 12 - 4 \times 1.5 = 12 - 6 = 6 \) cm

- Time (hours) = 5:
- Tall candle height: \( 16 - 5 \times 2.5 = 16 - 12.5 = 3.5 \) cm
- Short candle height: \( 12 - 5 \times 1.5 = 12 - 7.5 = 4.5 \) cm

- Time (hours) = 6:
- Tall candle height: \( 16 - 6 \times 2.5 = 16 - 15 = 1 \) cm
- Short candle height: \( 12 - 6 \times 1.5 = 12 - 9 = 3 \) cm

- Time (hours) = 7:
- Tall candle height: \( 16 - 7 \times 2.5 = 16 - 17.5 = -1.5 \) cm
- Short candle height: \( 12 - 7 \times 1.5 = 12 - 10.5 = 1.5 \) cm

### Completed Table

\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 & 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}

The table now shows how the heights of both candles change over a period of 7 hours.