We are given that the height of a candle decreases linearly over time. Here’s how to solve this:

1. Define the points: Since we know this relationship is linear, we can use the given heights at two different times to find the linear equation of the form $h(t) = mt + b$, where:

   • $h(t)$ is the height in cm at time $t$ in hours.
   • $m$ is the slope (rate of change of height over time).
   • $b$ is the y-intercept (initial height at $t = 0$).

2. Identify the points:

   • At $t = 13$ hours, $h = 19.1$ cm.
   • At $t = 31$ hours, $h = 13.7$ cm.

3. Calculate the slope (m): $m = \frac{h_2 - h_1}{t_2 - t_1} = \frac{13.7 - 19.1}{31 - 13} = \frac{-5.4}{18} = -0.3$ So, the slope $m = -0.3$.

4. Write the equation: Using the point-slope form $h - h_1 = m(t - t_1)$ with the point $(13, 19.1)$: $h - 19.1 = -0.3(t - 13)$ Expanding this gives: $h = -0.3t + 19.1 + 3.9 = -0.3t + 23$ So the linear equation is: $h(t) = -0.3t + 23$

5. Find the height at $t = 15$ hours: Substitute $t = 15$ into the equation: $h(15) = -0.3(15) + 23 = -4.5 + 23 = 18.5$

Answer: The height of the candle after 15 hours is 18.5 cm.

Would you like further details on any part of this process?

Here are some related questions:

1. How do you determine if a relationship is linear from given points?
2. How would the height change if the slope was positive instead of negative?
3. What would the candle height be at time $t = 0$ using this model?
4. How does a change in slope affect the rate at which the candle burns down?
5. How can we confirm that a model is accurate beyond the given points?

Tip: Remember, the slope in a linear model represents the rate of change. A negative slope here means the height is decreasing over time.