Question

Model and Solve
A 12-inch candle and an 18-inch candle are lit at the same time. The 12-inch candle burns












inches every hour. The 18-inch candle burns 3 inches every hour.
Question 1
Which equation will yield the time it takes for the candles to burn down to the same height?
Responses
A 12 +


x = 18 + 3x12 + 3 4 x = 18 + 3x
B 12 −


x = 18 − 3x12 − 3 4 x = 18 − 3x
C 12 +


x = 18 − 3x12 + 3 4 x = 18 − 3x
D 12 −


x = 18 + 3x12 − 3 4 x = 18 + 3x
Question 2
How much time will it take for the candles to burn down to the same height?
Responses
A 2 hours2 hours
B 2 hours and 20 minutes2 hours and 20 minutes
C 2 hours and 40 minutes2 hours and 40 minutes
D 3 hours

Answer 1

To model the situation with the candles, let's denote \( x \) as the time in hours after the candles are lit.

### Step 1: Create an equation for both candles

- The 12-inch candle burns at a rate of 1 inch per hour. So, after \( x \) hours, the height of the 12-inch candle will be:
\[
12 - 1x = 12 - x
\]

- The 18-inch candle burns at a rate of 3 inches per hour. So, after \( x \) hours, the height of the 18-inch candle will be:
\[
18 - 3x
\]

### Step 2: Set the heights equal to find when they are the same
To find out when the heights are equal, set the two equations equal to each other:

\[
12 - x = 18 - 3x
\]

### Step 3: Solve for \( x \)
Now, solve the equation:
\[
12 - x = 18 - 3x
\]

Add \( 3x \) to both sides:
\[
12 + 2x = 18
\]

Subtract 12 from both sides:
\[
2x = 6
\]

Dividing by 2:
\[
x = 3
\]

### Conclusion to Question 1
Now, let's analyze the responses:

Since we ended up with \( 12 - x = 18 - 3x \), the correct response is:

**D: \( 12 - x = 18 - 3x \)**

### Question 2: Finding the time
From our calculation, we found that \( x = 3 \). Therefore, it will take:

**D: 3 hours**

Let's summarize:

1. **Response to Question 1:** D \( 12 - x = 18 - 3x \)
2. **Response to Question 2:** D \( 3 \) hours