Nina lit a candle and measured its height after different lengths of time. After 0.5 hour, the height of the candle was 16.5 centimeters. After 1.5 hours, the height of the candle was 13.5 centimeters. Assume the relationship is linear. Find and interpret the rate of change. The rate of change is hour. Find and interpret the initial value. The initial value is

To analyze the relationship between the height of the candle and the time elapsed, we'll assume a linear relationship of the form: \[ h(t) = mt + b \] where: - \( h(t) \) is the height of the candle at time \( t \) (in centimeters), - \( m \) is the rate of change (slope), - \( b \) is the initial height of the candle (y-intercept). ### Given Data Points: 1. At \( t = 0.5 \) hours, \( h = 16.5 \) cm. 2. At \( t = 1.5 \) hours, \( h = 13.5 \) cm. ### 1. Finding the Rate of Change (\( m \)): The rate of change (slope) is calculated using the two points: \[ m = \frac{h_2 - h_1}{t_2 - t_1} = \frac{13.5\, \text{cm} - 16.5\, \text{cm}}{1.5\, \text{hr} - 0.5\, \text{hr}} = \frac{-3\, \text{cm}}{1\, \text{hr}} = -3\, \text{cm/hour} \] **Interpretation:** **The rate of change is –3 cm per hour.** *This means the candle is burning at a rate of 3 centimeters each hour.* ### 2. Finding the Initial Value (\( b \)): Use one of the data points to solve for \( b \). Let's use \( t = 0.5 \) hours and \( h = 16.5 \) cm: \[ 16.5 = (-3)(0.5) + b \\ 16.5 = -1.5 + b \\ b = 16.5 + 1.5 \\ b = 18\, \text{cm} \] **Interpretation:** **The initial value is 18 cm.** *This represents the height of the candle at the moment Nina lit it (time zero).* ### Summary: - **Rate of Change:** –3 cm per hour *The candle decreases in height by 3 centimeters every hour.* - **Initial Value:** 18 cm *The candle was 18 centimeters tall when it was first lit.*