Application problem with a linear function: Finding a coordinate given two points

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 13 hours of burning, a candle has a height of 21.1 centimeters. After 24 hours of burning, its height is 17.8 centimeters. What is the height of the candle after 23 hours?

Answer 1

The general formula for a linear function is:

y = mx + b

or

h(t) = m*t + b where m = slope and b = y-intercept

The point (t, h(t)) = (time in hours, height in centimeters)

Given the two points:

(13, 21.1)

(24,17.8)

The slope, m:

m = (y₂-y₁) / (x₂-x₁)

m = (17.8 - 21.1) / (24 - 13) = -3.3 / 11 = -0.3

h(t) = m*t + b

h(t) = -0.3*t + b

To determine b, pick one point and substitute it into this equation.

Let's use point (24, 17.8) for (t, h(t)).

h(t) = -0.3*t + b

17.8 = -0.3*(24) + b

b = 17.8 + 7.2 = 25

b = 25

So, the general linear formula is:

h(t) = -0.3*t + 25

At 23 hours:

h(23) = -0.3*23 + 25 = 18.1

Therefore, the height of the candle at 23 hours is 18.1 cm.

Since 18.1 cm is right in-between 21.1 cm and 17.8 cm, and quite close to 17.8 cm, the answer passes the reasonable test.