Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A moving company charges a minimum of $$\$250$$ for a move. An additional $$\$ 100$$ per hour is charged for time in excess of two hours. Write a function \(C(t)\) that gives the cost of a move that takes \(t\) hours to complete.

Step-by-Step Solution

Answer

The cost of a move, \(C(t)\), is given by the piecewise function: \(C(t)=\begin{cases} \$250 & \text{if } 0 \leq t \leq 2 \ \$250 + \$100(t - 2) & \text{if } t > 2 \end{cases}\), where \(t\) is the time in hours the move takes.

Step 1: Identify the function for the first 2 hours

For the first 2 hours, the cost remains constant at $250 regardless of the time. Therefore, \(C(t)=\$250\) when \( 0 \leq t \leq 2 \) hours.

Step 2: Identify the function for times more than 2 hours

For every hour more than 2, there is an additional charge of $100. Thus, any extra hour involves an additional $100 besides the constant $250 charge. Therefore, after 2 hours, the function would be \(C(t)=\$250 + \$100(t - 2)\), when \( t > 2 \) hours.

Step 3: Combine both functions into a piecewise function

Now, combine both functions into a piecewise function to generalize the pricing method. This function will include both parts corresponding to services below or over 2 hours: \(C(t)=\begin{cases} \$250 & \text{if } 0 \leq t \leq 2 \ \$250 + \$100(t - 2) & \text{if } t > 2 \end{cases}\).

Key Concepts

Calculus

Calculus is an essential branch of mathematics focusing on limits, functions, derivatives, integrals, and infinite series. In the context of piecewise functions, calculus helps describe scenarios where a function changes its rule based on the input, much like a moving company's charges change after a certain number of hours. These function rules can potentially be more complex and require differentiation to find rates of change, or integration to find total values over intervals. Calculus provides the tools for analyzing and understanding these piecewise-defined functions in a precise and structured manner.

Function Modeling

Function modeling involves creating equations to represent real-world scenarios. The exercise provided is a classic example, where a moving company's pricing structure is represented as a piecewise function. Models like these are widely used because they provide clarity and help in anticipating costs for services. When creating function models, it's crucial to define the relevant intervals and the specific rules that apply to those intervals to ensure that the model accurately reflects the situation at hand.

Step-by-Step Problem Solving

Step-by-step problem solving is a methodical approach to finding solutions to complex problems. By breaking down the process into smaller, more manageable steps, students can systematically work through the particulars of a problem. In the case of the moving company charges, the problem is addressed in a three-step process: identifying the flat rate cost, calculating additional costs after a certain time threshold, and then combining these into a coherent piecewise function. This structured approach helps students to not only find the solution but also understand the reasoning behind each step.

Rates of Change

Rates of change are a part of calculus that describe how a function changes as its input changes. In practical terms, it helps us understand how one quantity affects another in real-time. For instance, the additional charge per hour by the moving company after the first two hours represents a linear rate of change in the total moving cost with respect to time. This is particularly significant in understanding how costs accumulate over time and aids in budgeting and forecasting expenses.