Suppose that a household's monthly water bill (in dollars) is a linear function of the amount of water the household uses (in hundreds of cubic feet, HCF). When graphed, the function gives a line with a slope of 1.65 . See the figure below.
If the monthly cost for 18 HCF is $\mathrm{\$}46.96$, what is the monthly cost for $14\mathrm{HCF}$ ?

Step 1 ：Given that the monthly cost for 18 HCF is $46.96 and the slope of the line is 1.65, we can use the formula for a line, $y=mx+b$, where $m$ is the slope, $x$ is the independent variable (in this case, the amount of water used), and $b$ is the y-intercept (the base cost of water regardless of usage).

Step 2 ：First, we can solve for $b$ using the given information. Substituting the given values into the equation, we get $46.96=1.65\ast 18+b$. Solving for $b$, we find that $b=46.96-1.65\ast 18=17.26$.

Step 3 ：Next, we can use the value of $b$ to find the cost for 14 HCF. Substituting the values into the equation, we get $y=1.65\ast 14+17.26$. Solving for $y$, we find that the cost for 14 HCF is $40.36.

Step 4 ：Final Answer: The monthly cost for 14 HCF is $\boxed{{\displaystyle 40.36}}$ dollars.