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.