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 5 hours of burning, a candle has a height of 24 centimeters. After 19 hours of burning, it’s height is 18.4 centimeters. What is the height of the candle after 10 hours?

Answer 1

Step 1: Write out the general equation for a linear function

`\begin{gathered} \text{The general equation for a linear function is} \\ y=mx+c \\ \text{Where} \\ y=\text{dependent variable} \\ x=\text{independent variable} \\ m=\text{slope} \\ c=\text{intercept on the y-axis} \end{gathered}`

Step 2: Write out a linear function for the height and the amount of time

If h represents height and t represents time. The equation of the linear function connecting h and t would be

`h=mt+c`

Step 3: Write out the general formula for finding the equation of linear function given two different coordinates

`(y-y_1)/(x-x_1)=(y_2-y_1)/(x_2-x_1)`

For the function of h and t, the formula would be

`\begin{gathered} (h-h_1)/(t-t_1)=(h_2-h_1)/(t_2-t_1) \\ \text{where} \\ h_1=\text{initiah height=24cm} \\ h_2=\text{final height=18.4cm} \\ t_1=\text{initial time=5hours} \\ t_2=\text{final time=19hours} \end{gathered}`

Substitute for the given parameters in the formula

`(h-24)/(t-5)=(18.4-24)/(19-5)`
`\begin{gathered} (h-24)/(t-5)=(-5.6)/(14) \\ (h-24)/(t-5)=(-2)/(5) \\ \text{cross multiply} \\ 5(h-24)=-2(t-5) \\ 5h-120=-2t+10 \\ 5h=-2t+10+120 \\ 5h=-2t+130 \end{gathered}`

Divide through by 5

`\begin{gathered} (5h)/(5)=(-2t)/(5)+(130)/(5) \\ h=(-2t)/(5)+26 \end{gathered}`

The above function connect the height and the amount of time together

Step 4: Find the height of the candle after 10 hours

Hence, the height of the candle after 10 hours is 22cm