Question

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 22 centimeters. after 27 hours of burning, its height is 11.8 centimeters. what is the height of the candle after 16 hours?

Explanation:

Step1: Define linear function form

Let \( h(t) = mt + b \), where \( h(t) \) is height (cm), \( t \) is time (hours), \( m \) is slope, \( b \) is initial height.

Step2: Set up two equations

From given data:

1. \( 22 = 10m + b \)
2. \( 11.8 = 27m + b \)

Step3: Subtract equations to find \( m \)

Subtract equation 1 from equation 2:
$$11.8 - 22 = 27m + b - (10m + b)$$
$$-10.2 = 17m$$
$$m = \frac{-10.2}{17} = -0.6$$

Step4: Solve for \( b \)

Substitute \( m=-0.6 \) into \( 22 = 10m + b \):
$$22 = 10(-0.6) + b$$
$$22 = -6 + b$$
$$b = 22 + 6 = 28$$

Step5: Find height at 16 hours

Use \( h(t) = -0.6t + 28 \), substitute \( t=16 \):
$$h(16) = -0.6(16) + 28$$
$$h(16) = -9.6 + 28 = 18.4$$

Answer:

18.4 centimeters