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 7 hours of burning, a candle has a height of 22.9 centimeters. after 23 hours of burning, its height is 18.1 centimeters. what is the height of the candle after 22 hours? candle height (in cm) burning time (in hours) centimeters

Explanation:

Step1: Find the slope of the linear function

The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, \( x_1 = 7 \), \( y_1=22.9 \), \( x_2 = 23 \), \( y_2 = 18.1 \). So, \( m=\frac{18.1 - 22.9}{23 - 7}=\frac{- 4.8}{16}=- 0.3 \).

Step2: Use point - slope form to find the equation

The point - slope form of a line is \( y - y_1=m(x - x_1) \). Using the point \((7,22.9)\) and \( m=-0.3 \), we have \( y-22.9=-0.3(x - 7) \). Simplify it: \( y-22.9=-0.3x + 2.1 \), so \( y=-0.3x+25 \).

Step3: Find the height at \( x = 22 \) hours

Substitute \( x = 22 \) into the equation \( y=-0.3x + 25 \). Then \( y=-0.3\times22 + 25=-6.6 + 25 = 18.4 \). Or, since the time changes from 23 hours to 22 hours (a decrease of 1 hour), and the slope is - 0.3 (the height increases by 0.3 cm for each hour decrease in burning time). At 23 hours, the height is 18.1 cm. So at 22 hours, the height is \( 18.1+0.3 = 18.4 \) cm.

Answer:

\( 18.4 \)