Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

After 3 miles of dif cult climbing in the morning, a group of hikers has reached a plateau and they are con dent they can maintain a steady pace for the next 10 miles. After covering a total of 13 miles, they ll set camp. Twenty minutes after reaching the plateau, they ve covered \(1 \frac{1}{3}\) miles. Express the total daily mileage as a function of \(t .\) where \(t\) is the number of hours spent hiking since they reached the plateau. What is the domain of the function?

Step-by-Step Solution

Answer

The function representing the total daily mileage as a function of \(t\) is \(M(t) = 3 + 4t\) where \(0 \leq t \leq 2 \frac{5}{6}\).

Step 1: Expressing the initial mileage

Firstly, consider the initial condition where the group has already covered 3 miles. This will be the base value for the function. So, the function starts as \(M(t) = 3\).

Step 2: Calculating hours spent for the next phase of hiking

The question mentions that twenty minutes after reaching the plateau, they've covered \(1 \frac{1}{3}\) miles. This means in one hour (\(1 \frac{1}{3}\) of an hour), they would cover \(1 \frac{1}{3}\times 3 = 4\) miles. Now, this should be added to the function as a rate (\(4\) miles per hour), which leads us to \(M(t) = 3 + 4t\).

Step 3: Determining the domain of the function

The group plans to cover a total of 10 miles after reaching the plateau. Given their rate of \(4\) miles per hour, they would take \(10/4 = 2.5\) hours to cover that distance. Since they've already spent \(1/3\) of an hour covering the first \(1 \frac{1}{3}\) miles, the total time they will spend is \(2.5 + 1/3 = 2 \frac{5}{6}\) hours. Hence, the domain of \(t\) is \(0 \leq t \leq 2 \frac{5}{6}\).

Key Concepts

Hiking Problem

The hiking problem involves a group of adventurers who have already completed 3 miles of tough climbing before reaching a plateau. They're ready to continue their hike over a flatter terrain. The problem sets a scenario where they have 10 additional miles to travel over this steady pace before they stop to set camp. The main task is to express their total mileage as a function of time, represented by the variable \(t\), which stands for the number of hours they have been hiking on the plateau. By understanding the context and constraints, we can solve for how far they have traveled given any time \(t\). Understanding this hike requires not only math but also an interpretation of real-world travel, reminding us of real hikes where pace changes with terrain shifts. This type of problem combines an appreciation for nature with mathematical skills, illustrating how math models our real experiences.

Miles per Hour

In this scenario, the hikers travel at a rate of a certain number of miles per hour. They've already managed to cover \(1 \frac{1}{3}\) miles in twenty minutes, translating to a pace or speed at which they continue to travel. To calculate their speed in miles per hour, convert the twenty minutes into a fraction of an hour, which is \(\frac{1}{3}\) hour. Hence, being able to cover \(1 \frac{1}{3}\) miles in this time implies that in a full hour, they continue at the rate: - \(1 \frac{1}{3} \times 3 = 4\) miles per hourThis uniform rate allows us to confidently predict how much ground they can cover over time by multiplying \(4\) by the number of hours, \(t\). Understanding units and unit conversion is critical in solving problems like these, where different time measures need to be coherent.

Domain of a Function

The domain of a function refers to all possible input values the function can accept, in this case, it translates into how long the hikers can continue hiking. As they are on a plateau, the problem specifies they aim to cover an additional 10 miles. Since they travel at a consistent rate of 4 miles per hour, we can figure out how many hours they need to hike by dividing 10 miles by their speed: - \(\frac{10}{4} = 2.5\) hoursHowever, they've already spent \(\frac{1}{3}\) hour covering the initial \(1 \frac{1}{3}\) miles, leaving them with a total hiking time of: - \(2.5 + \frac{1}{3} = 2 \frac{5}{6}\) hoursThus, the domain of the function represents time \(t\) between 0 and \(2 \frac{5}{6}\) hours, capturing the full duration they can spend traversing on the plateau portion of their hike.

Rate of Change

The rate of change in this exercise is effectively synonymous with the speed at which the hikers travel. It describes how their mileage increases with time while hiking on the plateau. Here, the rate of change is given as 4 miles per hour.The formula \(M(t) = 3 + 4t\) summarizes how the initial 3 miles combine with any additional travel, as time \(t\) increases. It paints a picture of a steady and predictable journey, showing exactly how much mileage increases as each hour passes. Understanding rate of change is crucial in various real-world applications, from monitoring speed and fuel consumption while driving, to budgeting time during hikes like the one described. It underscores an essential principle of mathematics: connecting patterns, predictions and real-life events, reinforcing that math is not just theoretical but applied.