To determine the formula that best describes the cost of piano lessons given in the table, we need to consider the cost per hour and any potential fixed costs. The table provides four data points:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Hours of Instruction} & \text{Cost} \\
\hline
6 & \$230.00 \\
\hline
10 & \$330.00 \\
\hline
18 & \$530.00 \\
\hline
25 & \$705.00 \\
\hline
\end{array}
\][/tex]
To find the cost per hour and any fixed cost, we will look at the changes in cost with respect to the changes in hours. We will calculate the increase in cost for each increase in hours and use this to find a consistent cost per hour.
First, consider the increase in cost from [tex]\(6\)[/tex] to [tex]\(10\)[/tex] hours:
[tex]\[
\frac{\$330 - \$230}{10 - 6} = \frac{\$100}{4} = \$25 \text{ per hour}
\][/tex]
Assuming the cost per hour is consistent, let's determine if all other points follow this rate.
Next, check the increase in cost from [tex]\(10\)[/tex] to [tex]\(18\)[/tex] hours:
[tex]\[
\frac{\$530 - \$330}{18 - 10} = \frac{\$200}{8} = \$25 \text{ per hour}
\][/tex]
Finally, check the increase in cost from [tex]\(18\)[/tex] to [tex]\(25\)[/tex] hours:
[tex]\[
\frac{\$705 - \$530}{25 - 18} = \frac{\$175}{7} = \$25 \text{ per hour}
\][/tex]
Seeing that the cost per hour is indeed \[tex]$25 consistently, let's check if there's any fixed cost.
From the first data point, for 6 hours at \$[/tex]25 per hour:
[tex]\[
6 \times \$25 = \$150
\][/tex]
However, the cost given is \[tex]$230, which means there might be a fixed cost:
\[
\$[/tex]230 - \[tex]$150 = \$[/tex]80
\]
Thus, the total cost formula seems to be:
[tex]\[
\text{Cost} = \$25 \times \text{hours} + \$80
\][/tex]
This matches one of the given options:
[tex]\[
\textbf{C. } \$ 25 \text{ per hour of instruction, plus } \$ 80
\][/tex]
Thus, the best formula that describes the cost of piano lessons is:
[tex]\[
\boxed{\$ 25 \text{ per hour of instruction, plus } \$ 80}
\][/tex]
