- Identify the problem as a linear relationship of the form \(P(t) = mt + b\).
- Calculate the hourly fee \(m\) using the slope formula \(m = \frac{P_2 - P_1}{t_2 - t_1}\) with the provided data points.
- Verify the slope is consistent across different pairs of points.
- The hourly rate is calculated to be \(\boxed{150}\) dollars per hour.
Explanation
- Analyzing the Linear Relationship
The problem states that the total cost \(P\) of renting a music studio is related to the number of hours \(t\) by a linear function, which can be expressed in the form \(P(t) = mt + b\). Here, \(m\) represents the hourly fee (the slope) and \(b\) represents the initial fee (the y-intercept). We are provided with a table of values: \((5, 1050)\), \((7, 1350)\), and \((8.5, 1575)\). Our objective is to determine the hourly fee \(m\).
- Calculating the Slope
A linear function has a constant rate of change, which is the slope \(m\). We can calculate this rate of change between any two points \((t_1, P_1)\) and \((t_2, P_2)\) using the formula:
\[m = \frac{P_2 - P_1}{t_2 - t_1}.\]
Using the first two data points, \((5, 1050)\) and \((7, 1350)\):
\[m = \frac{1350 - 1050}{7 - 5} = \frac{300}{2} = 150.\]
Thus, the hourly fee is 150 dollars.
- Verifying the Hourly Fee
To ensure consistency, we verify the slope using the second and third data points, \((7, 1350)\) and \((8.5, 1575)\):
\[m = \frac{1575 - 1350}{8.5 - 7} = \frac{225}{1.5} = 150.\]
Since both calculations yield the same result, the hourly fee for the studio is constant at 150 dollars per hour.
- Conclusion
We have determined that the additional price per hour of using the studio is 150 dollars. Therefore, the final answer is \(\boxed{150}\).
Examples
Understanding linear relationships is essential in budgeting for services with base fees, such as studio rentals, utility bills, or specialized taxi services. For instance, if you are planning to record a podcast, knowing how the base fee and hourly rate combine helps you accurately estimate your total production costs based on the number of hours you book. By mastering this concept, you can effectively manage project budgets and compare pricing plans across different vendors.