Question

A truck leaves Regina and drives eastbound. Due to road construction, the truck takes \(2 \mathrm{h}\) to travel the first \(80 \mathrm{km} .\) Once it leaves the construction zone, the truck travels at \(100 \mathrm{km} / \mathrm{h}\) for the rest of the trip. a) Let \(v\) represent the average speed, in kilometres per hour, over the entire trip and \(t\) represent the time, in hours, since leaving the construction zone. Write an equation for \(v\) as a function of \(t\) b) Graph the function for an appropriate domain. c) What are the equations of the asymptotes in this situation? Do they have meaning in this situation? Explain. d) How long will the truck have to drive before its average speed is \(80 \mathrm{km} / \mathrm{h} ?\) e) Suppose your job is to develop GPS technology. How could you use these types of calculations to help travellers save fuel?

Short answer

a) \( v = \frac{80 + 100t}{2 + t} \). b) Graph \( v = \frac{80 + 100t}{2 + t} \). c) Horizontal asymptote \( y = 100 \). d) \( t = 4 \) hours. e) Optimizes routes for fuel and time efficiency.

Step-by-step solution

Find initial conditions

The truck takes 2 hours to travel 80 km in the construction zone. Thus, the initial average speed for the first 80 km is determined by dividing the distance by the time:\[\frac{80 \text{ km}}{2 \text{ h}} = 40 \text{ km/h}\]

Determine the function for total distance

After leaving the construction zone, the truck travels at 100 km/h for time \( t \) in hours. Therefore, the total distance \( D \) after traveling for time \( t \) can be expressed as: \[ D = 80 +(100t) \]

Establish the equation for total time

The total time taken for the trip, including time in the construction zone and time \( t \) after leaving the construction zone, is: \[T = 2 + t \]

Calculate the average speed function

The average speed \( v \) is given by the total distance divided by the total time. Using our previous expressions for distance and time: \[ v = \frac{D}{T} = \frac{80 + 100t}{2 + t} \]

Graph the function

Plot the function \( v = \frac{80 + 100t}{2 + t} \) against \( t \) for an appropriate domain. Since the truck starts traveling at time \( t = 0 \), choose a domain such as \( 0 \leq t \leq 10 \). The graph should depict how the average speed increases asymptotically as \( t \) increases.

Identify and interpret the asymptotes

The horizontal asymptote can be found by considering the behavior of \( v \) as \( t \) approaches infinity: \[ \lim_{t \to \infty} \frac{80 + 100t}{2 + t} = 100 \text{ km/h} \]This means the average speed will approach 100 km/h as the time increases.

Calculate time to achieve specific average speed

To achieve an average speed of 80 km/h, set \( v = 80 \) and solve for \( t \): \[ 80 = \frac{80 + 100t}{2 + t} \]Multiply both sides by \( 2 + t \) and solve the resulting equation for \( t \): \[ 80(2 + t) = 80 + 100t \] \[ 160 + 80t = 80 + 100t \] \[ 160 - 80 = 20t \] \[ t = 4 \text{ hours} \]

Discuss the application for GPS technology

Understanding such calculations can help design GPS systems that inform travelers about optimal speeds and routes to minimize travel time and fuel consumption. By predicting average speed and travel time accurately, GPS technology can optimize fuel efficiency and reduce journey durations.

Key concepts

Distance-Time Relationship

Understanding the distance-time relationship is crucial in solving problems involving motion. Here, we have a truck that spends 2 hours traveling through a construction zone at an average speed of 40 km/h to cover 80 km. After leaving the construction zone, it continues at 100 km/h.
To calculate the total distance covered over time, we can use the formula:

• Total distance before construction zone: 80 km
• Additional distance after the construction zone: 100t km

Putting these two together, we get the total distance function:
\[ D = 80 + 100t \] This function tells us how far the truck has traveled as a function of time, which provides a solid foundation for further calculations in the exercise.

Average Speed Function

The average speed of the truck is a crucial piece of information. It tells us how fast, on average, the truck is moving over the entire trip, considering both the construction and non-construction periods.
To find this, we need both the total distance and the total time taken for the trip:

• Total distance: \( D = 80 + 100t \)
• Total time: \( T = 2 + t \)

The formula for average speed is:
\[ v = \frac{D}{T} \] Substituting the expressions for \( D \) and \( T \), we get the average speed function:
\[ v = \frac{80 + 100t}{2 + t} \] This function will help us not only understand how the average speed changes over time but also graph it and identify any asymptotes.

Graphing Functions

Graphing the function \( v = \frac{80 + 100t}{2 + t} \) helps visualize how the truck's average speed evolves over time. Start by choosing an appropriate domain, such as \( 0 \leq t \leq 10 \).
When you plot the function, you'll notice the average speed increasing asymptotically. The graph will help show how the initial low average speed (due to the slow down in the construction zone) eventually approaches a higher value as time goes on.
Graphs are vital since they provide a visual representation, making it easier to understand how quickly or slowly average speed changes over time.

Asymptotes in Speed Calculations

Asymptotes are lines that a curve approaches but never actually touches or crosses. In our average speed function \( v = \frac{80 + 100t}{2 + t} \), the average speed approaches 100 km/h as time increases indefinitely.
The horizontal asymptote here can be derived by evaluating the limit of the function as time approaches infinity:
\[ \lim_{t \to \infty} \frac{80 + 100t}{2 + t} = 100 \text{ km/h} \] This tells us that, after driving for a very long period and leaving behind the initial 2-hour delay, the truck's average speed will nearly equal its constant speed of 100 km/h.
This insight is pivotal, as it explains that even with initially lower average speeds (due to the construction zone), the truck will eventually reach its cruising speed.