A candle is 15 inches tall after burning for 3 hours. After 5 hours, it is 13 inches tall.
Write a linear equation to model the relationship between height h of the candle and time t.
Predict how tall the candle will be after burning for 8 hours.
To write a linear equation to model the relationship between the height h of the candle and time t, we can use the slope-intercept form of a linear equation, y = mx + b, where y is the height of the candle, x is the time in hours, m is the slope (rate of change of height with respect to time), and b is the y-intercept (initial height of the candle).
First, let's calculate the rate of change of height with respect to time:
We are given that the candle is 15 inches tall after burning for 3 hours and 13 inches tall after burning for 5 hours.
The change in height, Δh, is given by: Δh = h2 - h1, where h1 is the height at the initial time and h2 is the height at a later time.
Δh = 13 - 15 = -2 inches (negative because height decreases)
The change in time, Δt, is given by: Δt = t2 - t1, where t1 is the initial time and t2 is the later time.
Δt = 5 - 3 = 2 hours
The rate of change of height with respect to time, m, is given by: m = Δh/Δt
m = -2/2 = -1 inches per hour
Now that we have the slope, let's determine the y-intercept, b:
We are given that the candle is 15 inches tall after burning for 3 hours.
Using the slope-intercept form, we can substitute the initial height (h) and initial time (t) into the equation: h = mt + b
15 = -1(3) + b
15 = -3 + b
b = 15 + 3
b = 18
So the linear equation to model the relationship between the height h of the candle and time t is:
h = -t + 18
To predict how tall the candle will be after burning for 8 hours, we can substitute t = 8 into the equation:
h = -8 + 18
h = 10
Therefore, the candle is predicted to be 10 inches tall after burning for 8 hours.