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 11 hours of burning, a candle has a height of 16.5 centimeters. After 30 hours of burning, its height is 7 centimeters. What is the height of the candle after hours?
height (y) is dependent on burning time (x)
it loses 9.5 cm (16.5 - 7) over 19 hr (30 - 11)
... so the slope of the line is ... -9.5 / 19 = -1/2
y = - 1/2 x + b ... substitute one of the points to find b (the y-intercept)
To find the height of the candle after a specific number of hours, we need to determine the equation of the linear function that represents the relationship between the height and the time.
Let's assign the variable h to represent the height of the candle and t to represent the time in hours. According to the given information, we have two data points: (11, 16.5) and (30, 7).
To find the equation of a linear function, we need to determine the slope (m) and the y-intercept (b).
First, let's calculate the slope using the formula:
m = (y2 - y1) / (x2 - x1)
Using the points (11, 16.5) and (30, 7):
m = (7 - 16.5) / (30 - 11)
m = -9.5 / 19
m = -0.5
Next, we can find the y-intercept (b) by substituting one of the points into the equation:
16.5 = -0.5 * 11 + b
Simplifying the equation:
16.5 = -5.5 + b
b = 16.5 + 5.5
b = 22
Now that we have the slope (m = -0.5) and the y-intercept (b = 22), we can write the equation of the linear function:
h = mt + b
h = -0.5t + 22
To find the height of the candle after a specific number of hours, substitute that value into the equation.
For example, if we want to find the height after 50 hours, we can substitute t = 50 into the equation:
h = -0.5(50) + 22
h = -25 + 22
h = -3
Therefore, the height of the candle after 50 hours would be -3 centimeters.