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 5 hours of burning, a candle has a height of 26.5 centimeters. After 24 hours of burning, its height is 24.6 centimeters. What is the height of the candle after 11 hours?
Don't forget your Algebra I now that you're taking calculus.
You have two points: (5,26.5) and (24,24.6)
So figure the slope of the line, and use the point-slope form of the equation.
Then find f(11)
Or, just do linear interpolation.
11 is 6/19 of the way from 5 to 24
so, the height will be 6/19 of the way from 26.5 to 24.6
To solve this problem, we need to find the slope-intercept form of the linear function that represents the height of the candle as a function of time.
Let's use the formula for a linear equation: y = mx + b, where y is the height of the candle, x is the time in hours, m is the slope, and b is the y-intercept.
From the given information, we have two data points: (5, 26.5) and (24, 24.6). We can use these points to find the slope, m:
m = (y₂ - y₁) / (x₂ - x₁)
m = (24.6 - 26.5) / (24 - 5)
m = -1.9 / 19
m = -0.1
Now that we know the slope, we can substitute one of the data points into the equation to solve for b:
26.5 = -0.1(5) + b
26.5 = -0.5 + b
b = 27
So, the equation for the height of the candle as a function of time is:
y = -0.1x + 27
To find the height of the candle after 11 hours, we substitute x = 11 into the equation:
y = -0.1(11) + 27
y = -1.1 + 27
y = 25.9
Therefore, the height of the candle after 11 hours is 25.9 centimeters.