Answer :
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]
### 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]