
Suppose that the height (in centimeters) of a candle is a linear function of the amount of time (in hours) it has been burning. After 10 hours of burning, a candle has a height of 24 centimeters. After  21 hours of burning, its height is 21.8 centimeters. What is the height of the candle after 19 hours?


Let’s solve this problem by modeling the height of the candle as a linear function of time. We can express this linear relationship as:

\[
h(t) = mt + b
\]

where:
- \( h(t) \) is the height of the candle after \( t \) hours,
- \( m \) is the rate at which the candle height decreases per hour,
- \( b \) is the initial height of the candle when \( t = 0 \).

### Step 1: Set up equations for given data points

We know:
1. After 10 hours, the height is 24 cm:
   \[
   h(10) = 24
   \]
2. After 21 hours, the height is 21.8 cm:
   \[
   h(21) = 21.8
   \]

Substituting these into the equation \( h(t) = mt + b \):

1. \( 24 = 10m + b \)
2. \( 21.8 = 21m + b \)

### Step 2: Solve the system of equations

To find \( m \) and \( b \), subtract the first equation from the second:

\[
21.8 - 24 = (21m + b) - (10m + b)
\]
\[
-2.2 = 11m
\]
\[
m = -\frac{2.2}{11} = -0.2
\]

Now substitute \( m = -0.2 \) into one of the original equations to find \( b \). Using the first equation:

\[
24 = 10(-0.2) + b
\]
\[
24 = -2 + b
\]
\[
b = 26
\]

Thus, the equation for the candle's height is:

\[
h(t) = -0.2t + 26
\]

### Step 3: Find the height after 19 hours

Now, substitute \( t = 19 \) into the equation:

\[
h(19) = -0.2(19) + 26
\]
\[
h(19) = -3.8 + 26
\]
\[
h(19) = 22.2 \, \text{cm}
\]

### Answer
The height of the candle after 19 hours is **22.2 centimeters**.

Would you like further details or have any questions?

Here are five related questions for further practice:
1. What would the height be after 5 hours?
2. How long will it take for the candle to reach a height of 20 cm?
3. At what rate is the candle height decreasing per hour?
4. What would the initial height of the candle be?
5. How long will it take for the candle to completely burn out (height reaches zero)?

**Tip:** In linear function problems, try identifying two points to calculate the slope, then use one point to find the y-intercept for quick equation setup.

Suppose that the height (in centimeters) of a candle is a linear function of the amount of time (in hours) it has been burning. After 10 hours of burning, a candle has a height of 24 centimeters. After 21 hours of burning, its height is 21.8 centimeters. What is the height of the candle after 19 hours?

Learn how to determine the height of a candle after 19 hours of burning by modeling the problem as a linear function. This solution uses algebraic techniques, including the slope-intercept formula, to establish a relationship between time and height. Perfect for students in Grades 8-10.

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