Problem 107 - Precalculus Essentials

Question

You will be developing functions that model given conditions. You commute to work a distance of 40 miles and return on the same route at the end of the day. Your average rate on the return trip is 30 miles per hour faster than your average rate on the outgoing trip. Write the total time, \(T,\) in hours, devoted to your outgoing and return trips as a function of your rate on the outgoing trip, \(x .\) Then find and interpret \(T(30) .\) Hint: Time traveled \(=\frac{\text { Distance traveled }}{\text { Rate of travel }}\)

Step-by-Step Solution

Answer

The total time function is \(T(x) = \frac{40}{x} + \frac{40}{x+30}\). When evaluated at \(x=30\), the result is \(T(30) = 2\) hours. This means that if you drive at 30 mph on the outgoing trip and speed up by 30 mph on the return trip, the total time spent commuting is 2 hours.

Step 1: Set Up the Equation for Outgoing and Return Trips

The first step is to write separate equations for the outgoing and return trips. The distance for both trips is 40 miles. If \(x\) is the speed on the outgoing trip, then the time spent would be \(\frac{40}{x}\). On the return trip, the speed is 30 mph faster, that is \(x+30\), thus the time spent would be \(\frac{40}{x+30}\).

Step 2: Write the Total Time Function

The total time function \(T(x)\) is the sum of the time spent on the outgoing and return trips. So, we have \(T(x) = \frac{40}{x} + \frac{40}{x+30}\).

Step 3: Evaluate the Function at \(T(30)\)

To find \(T(30)\), we replace \(x\) in the function with 30. That gives us \(T(30) = \frac{40}{30} + \frac{40}{60} = \frac{4}{3} + \frac{2}{3} = 2\).

Key Concepts

Function Evaluation

In mathematics, function evaluation is a critical process in which we find the output of a function given its input value.
Let's take a closer look at what this means through the context of our commuting problem. Here, we are asked to develop a function that models the total time spent on a commute based on the rate of travel on the outgoing trip. This function symbolically represents the time taken to travel a certain distance at a varying speed.

To evaluate the function for a specific rate of travel, say 30 miles per hour, you substitute the variable in the function with this value. Following the exercise, we plugged in the number 30 for the variable 'x' within the function to compute 'T(30)'. Understanding this substitution method is essential for function evaluation as it allows us to calculate specific instances and gain concrete information from abstract function definitions.

Rate of Travel

The rate of travel, often measured in units like miles per hour (mph), is a fundamental concept in both math and the real world. It represents the speed at which an object is moving.
In our exercise, the rate of travel varies between the outgoing trip and the return trip. We are given the rate of the return trip as 30 mph faster than the outgoing trip. If the outgoing rate is labeled as 'x', this means the return trip rate will be 'x + 30'. By clearly understanding the relationship between distance, rate, and time, embodied by the formula 'time = distance ÷ rate,' we can easily set up equations that model real-life situations.

By calculating these rates correctly, we can make informed decisions, like how early to leave for an appointment based on expected traffic speeds or determine how a change in speed affects overall travel time.

Time-Distance Relationship

The time-distance relationship is a crucial element in the study of motion and transportation. It connects the distance traveled with the time it takes to cover that distance at a certain rate.
In our example, the distance is constant at 40 miles, but the time varies depending on the rate of travel. Remember, as the rate increases, the time decreases, assuming distance remains the same. This inverse relationship helps us to craft a function that tells us the total commuting time.

By adding the time it takes to go to work and the time to return, which are both dependent on the respective rates, we can model the overall time spent commuting. Understanding this relationship is particularly important when planning trips, scheduling deliveries, or just getting through our daily routines efficiently.