Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

On the first part of a 280 -mile trip, a salesperson averaged 63 miles per hour. The salesperson averaged only 54 miles per hour on the last part of the trip because of an increased volume of traffic. (a) Write the total time \(t\) for the trip as a function of the distance \(x\) traveled at an average speed of 63 miles per hour. (b) Use a graphing utility to graph the time function. What is the domain of the function? (c) Approximate the number of miles traveled at 63 miles per hour when the total time is 4 hours and 45 minutes.

Step-by-Step Solution

Answer

The time function is \(t(x) = x/63 + (280 - x)/54\). The domain of this function is \(0 \leq x \leq 280\). After about 4 hours 45 minutes, roughly 189 miles would have been covered at a pace of 63 miles per hour.

Step 1: Create the Time Function 't'

Start by setting up an equation that represents the total distance travelled, bearing in mind that time is calculated as distance divided by speed. In this case, for the part of the trip travelled at 63 miles per hour, the distance covered will be \(x\), and the corresponding time will be \(x/63\). For the remaining part of the trip, which was travelled at 54 miles per hour, and covered a distance of \(280 - x\) miles, the corresponding time will be \((280 - x)/54\). Add both time expressions to get the total time \(t\) such that \(t(x) = x/63 + (280 - x)/54\).

Step 2: Graph the Function 't'

While it's not entirely clear what tool should be used for graphing the function, students are often familiar with several graphing utilities, such as graphing calculators, or online tools like Desmos or GeoGebra. The domain of the function will be \(0 \leq x \leq 280\) since \(x\) represents distance, which can't be negative, and can't exceed the total journey of 280 miles.

Step 3: Determining Miles Traveled

To approximate the number of miles traveled at 63 miles per hour when the total time is 4 hours and 45 minutes, one needs to substitute \(t = 4.75\) (as 45 minutes is 0.75 of an hour) into the total time function: \(4.75 = x/63 + (280 - x)/54\). Proceed by solving this equation for \(x\) yields approximatly \(x \approx 189\) miles was travelled with a speed of 63 miles per hour when total time was 4 hours and 45 minutes.

Key Concepts

Average Speed

Understanding average speed is key to solving time-distance problems. It helps determine how much time you spend traveling over a certain distance. To find the average speed, remember that it is the total distance traveled divided by the total time taken. In this exercise, the salesperson traveled a total of 280 miles but at different speeds. On the first part of the trip, the speed was 63 miles per hour. On the second part, where traffic slowed them down, it was 54 miles per hour.
When solving problems of this type, you would set up a time function to see how the average speeds contribute to the total time. This helps in finding out how much time each section of the journey took based on the speeds used. All of this is captured in the time-distance equation, just like the one setup in this exercise.

Algebraic Function

An algebraic function is a way to represent relationships and solve problems in algebra. In this exercise, we set up a function for the total time of the trip based on the distance traveled at different speeds. The function, \(t(x) = \frac{x}{63} + \frac{280 - x}{54}\), captures how long it took to travel the distances at each speed. This equation involves simple algebra, where \(x\) is the distance traveled at 63 miles per hour, and \(280 - x\) is the distance at 54 miles per hour.
The beauty of algebraic functions is that they bring clarity to otherwise complex time-speed-distance problems. By plugging in different values of \(x\), you can determine how they affect the total time. This way, the equation helps answer vital questions related to the trip.

Graphing Utility

Graphing utilities are very helpful in visualizing algebraic functions, especially in time-distance problems. In this context, we can use tools like Desmos or GeoGebra. These tools allow us to input the function \(t(x) = \frac{x}{63} + \frac{280 - x}{54}\) and visually explore how changes in distance, \(x\), impact total travel time, \(t\).
Using a graphing utility also helps in observing the domain, which for this problem is \(0 \leq x \leq 280\). This is because \(x\) can range from 0 (not traveling at all on the first part) to 280 miles (traveling the entire trip at 63 mph). Being able to see these relationships on a graph provides a clearer understanding of the dynamics between speed, distance, and time.

Distance Equation

The distance equation is a fundamental part of solving time-distance problems. In this exercise, the equation \(t(x) = \frac{x}{63} + \frac{280 - x}{54}\) represents the total travel time as a function of the distance \(x\) traveled at the first speed. This equation combines two pieces: the first part of the trip's time, \(\frac{x}{63}\), and the second part's time, \(\frac{280 - x}{54}\), demonstrating how different rates affect the total traveling time.
Understanding how to construct and use distance equations allows you to solve for unknowns, like determining the distance traveled at a specific average speed. To find how far the salesperson traveled at 63 mph in 4.75 hours, you'd solve \(4.75 = \frac{x}{63} + \frac{280 - x}{54}\). This gives you valuable insights into optimizing routes, making targeted travel plans, and understanding travel dynamics.