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 $\$ 46.96$, 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 \(46.96 = 1.65 * 18 + b\). Solving for \(b\), we find that \(b = 46.96 - 1.65 * 18 = 17.26\).

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 \(y = 1.65 * 14 + 17.26\). 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 \(\boxed{40.36}\) dollars.