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The problem states that Liam rented a pedal board for 5.5 hours and paid $93.75, and it asks to model the cost of renting the pedal board with an equation in point-slope form, which relates the cost of renting for $ x $ hours. The rental cost per hour is given as $12.50.

Here’s how we solve it step by step:

1. **Point-slope form of a line** is given by:
   \[
   y - y_1 = m(x - x_1)
   \]
   where $ m $ is the slope (rate of change), $ (x_1, y_1) $ is a point on the line (here, the number of hours and total cost), and $ y $ represents the total cost.

2. The slope $ m $ is the rate of renting the pedal board, which is $ 12.50 $ dollars per hour.

3. Liam rented the board for 5.5 hours and paid $93.75, so $ x_1 = 5.5 $ and $ y_1 = 93.75 $.

4. Substituting into the point-slope formula:
   \[
   y - 93.75 = 12.50(x - 5.5)
   \]

This equation models the cost of renting the pedal board.

To find the **one-time service charge**, you can solve for $ y $ when $ x = 0 $ (the cost when no hours are rented). Let’s calculate that:

\[
y - 93.75 = 12.50(0 - 5.5)
\]
\[
y - 93.75 = 12.50(-5.5) = -68.75
\]
\[
y = 93.75 - 68.75 = 25.00
\]

So, the one-time service charge is $25.

Let me know if you need further clarification or details! Here are some related questions:

1. How can you rewrite the equation in slope-intercept form?
2. What does the slope represent in this context?
3. What would the total cost be for renting the pedal board for 8 hours?
4. How does the point-slope form help in finding the one-time service charge?
5. Can this method be applied to any rental service with a fixed hourly rate?

**Tip:** Point-slope form is useful when you know one point and the rate of change—it easily lets you model real-world problems like this one!

Liam rented a pedal board for 5.5 hours and paid a total of $93.75. What is an equation in point-slope form that models the cost of renting a pedal board for x hours? How can Liam use the equation to find the one-time service charge?

Learn how to model the cost of renting a pedal board using point-slope form. This problem involves calculating the total cost based on rental time and solving for the one-time service charge. Suitable for students in Grades 8-10.

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