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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) & [tex]$16$[/tex] cm candle height [tex]$(cm)$[/tex] & [tex]$12$[/tex] cm candle height [tex]$(cm)$[/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}

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

A. [tex]$T = 16 - 2.5h$[/tex]
B. [tex]$T = 16 + 2.5h$[/tex]
C. [tex]$T = -16 + 2.5h$[/tex]
D. [tex]$T = -16 - 2.5h$[/tex]

Sagot :

GinnyAnsweridn
Alright, let's tackle Cara's question step by step by completing the table and identifying the function for the taller candle.

### Step 1: Understanding the candles' initial height and burn rates

- Tall candle: Initial height of 16 cm, burns at 2.5 cm per hour.
- Short candle: Initial height of 12 cm, burns at 1.5 cm per hour.

### Step 2: Using algebra to find the candle heights over time

To find the height of each candle at different times, we use the following equations:

- For the tall candle:
\( T = 16 - 2.5h \)
- For the short candle:
\( S = 12 - 1.5h \)

Here, \( h \) represents the time in hours. We can use these equations to fill in the table.

### Step 3: Filling out the table

#### Time: 0 hours
- Tall candle: \( 16 - 2.5(0) = 16 \) cm
- Short candle: \( 12 - 1.5(0) = 12 \) cm

#### Time: 1 hour
- Tall candle: \( 16 - 2.5(1) = 13.5 \) cm
- Short candle: \( 12 - 1.5(1) = 10.5 \) cm

#### Time: 2 hours
- Tall candle: \( 16 - 2.5(2) = 11 \) cm
- Short candle: \( 12 - 1.5(2) = 9 \) cm

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

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

#### Time: 5 hours
- Tall candle: \( 16 - 2.5(5) = 3.5 \) cm
- Short candle: \( 12 - 1.5(5) = 4.5 \) cm

#### Time: 6 hours
- Tall candle: \( 16 - 2.5(6) = 1 \) cm
- Short candle: \( 12 - 1.5(6) = 3 \) cm

#### Time: 7 hours
- Tall candle: \( 16 - 2.5(7) = -1.5 \) cm (below zero)
- Short candle: \( 12 - 1.5(7) = 1.5 \) cm

### Step 4: Completing the table

[tex]\[ \begin{array}{|c|c|c|} \hline \text{Time (hours)} & \text{16 cm candle height (cm)} & \text{12 cm candle height (cm)} \\ \hline 0 & 16 & 12 \\ \hline 1 & 13.5 & 10.5 \\ \hline 2 & 11.0 & 9.0 \\ \hline 3 & 8.5 & 7.5 \\ \hline 4 & 6.0 & 6.0 \\ \hline 5 & 3.5 & 4.5 \\ \hline 6 & 1.0 & 3.0 \\ \hline 7 & -1.5 & 1.5 \\ \hline \end{array} \][/tex]

### Step 5: Function in slope-intercept form for the tall candle

The equation for the height \( T \) of the taller candle in terms of the number of hours it has burned \( h \) is given by:

[tex]\[ T = 16 - 2.5h \][/tex]

### Conclusion

Thus, the function for the height of the taller candle is \( T = 16 - 2.5h \).

Therefore, the correct answer is:
[tex]\[ T = 16 - 2.5h \][/tex]
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