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 find the rate of change, we can use the formula for the slope \( m \) of a line, which is given by: \[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} = \frac{{13.5 - 16.5}}{{1.5 - 0.5}} = \frac{{-3}}{{1}} = -3 \] This means that the candle height decreases by 3 centimeters for every hour, indicating that as time passes, the candle burns down steadily. Now, to find the initial value, which represents the height of the candle at the start (when \( x = 0 \)), we can use the slope-intercept form. The height after 0.5 hours is 16.5 cm, so we can work out the initial value by extending back. Using the slope \( m = -3 \): At \( x = 0.5 \), \( y = 16.5 \): \[ y - 16.5 = -3(x - 0.5) \] Setting \( x = 0 \): \[ y - 16.5 = -3(0 - 0.5) \Rightarrow y - 16.5 = 1.5 \Rightarrow y = 18 \] Thus, the initial height of the candle is 18 centimeters, indicating how tall it was before being lit.