Question

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
1.55. See the figure below.
If the monthly cost for 18 HCF is $45.16, what is the monthly cost for 13 HCF?

Answer 1

To find the monthly cost for 13 HCF using the linear function given, we start by establishing the linear equation for the water bill. We know that the slope of the line is \(1.55\) and that the cost for \(18\) HCF is \(45.16\).

The linear equation can be represented in the form:

\[
C = mH + b
\]

where
- \(C\) is the cost,
- \(H\) is the amount of water used (in HCF),
- \(m\) is the slope (which is \(1.55\)),
- and \(b\) is the y-intercept (fixed cost).

We can now plug in the known value for \(H\) and \(C\) to find \(b\).

Using \(H = 18\) and \(C = 45.16\):

\[
45.16 = 1.55 \times 18 + b
\]

Calculating \(1.55 \times 18\):

\[
1.55 \times 18 = 27.9
\]

Now substitute this value back into the equation:

\[
45.16 = 27.9 + b
\]

To find \(b\), we rearrange the equation:

\[
b = 45.16 - 27.9 = 17.26
\]

Now that we have \(b\), we can write the full linear equation for the cost, which is:

\[
C = 1.55H + 17.26
\]

Now we can use this equation to find the cost for \(H = 13\):

\[
C = 1.55 \times 13 + 17.26
\]

Calculating \(1.55 \times 13\):

\[
1.55 \times 13 = 20.15
\]

Now substitute this back into the equation:

\[
C = 20.15 + 17.26 = 37.41
\]

Thus, the monthly cost for \(13\) HCF is:

\[
\boxed{37.41} \text{ dollars.}
\]