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Chapter 5: Problem 13

Ellie was talking with her grandmother about a trip she took this summer. Ellie made the trip in \(2.5 \mathrm{~h}\) traveling at \(65 \mathrm{mi} / \mathrm{h}\). Ellie's grandmother remembers that she made the same trip in about \(6 \mathrm{~h}\) when she was Ellie's age. (ht) a. What speed was Ellie's grandmother traveling when she made the trip? b. Explain how this is an application of inverse variation.

Short Answer

Expert verified
a. Grandmother's speed was about 27.08 mi/h. b. It demonstrates inverse variation as longer time leads to lower speed for constant distance.

Step by step solution

01

Identify Total Distance Traveled by Ellie

To find out how far Ellie traveled, we use the formula for distance: \[ \text{Distance} = \text{Speed} \times \text{Time} \]. Here, Ellie's speed is 65 mi/h and the time is 2.5 hours. So, the distance she traveled is \[ 65 \times 2.5 = 162.5 \text{ miles} \].
02

Calculate Grandmother's Speed

Now that we know the distance, we can find out the speed of Ellie's grandmother during her trip. Using the formula for speed: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]. We substitute the distance (162.5 miles) and her travel time (6 hours). \[ \text{Speed} = \frac{162.5}{6} \approx 27.08 \text{ mi/h} \].
03

Understand Inverse Variation

This situation illustrates inverse variation because as the time taken to travel increases, the speed decreases to cover the same distance. The product of speed and time remains constant (the total distance of 162.5 miles in this case).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distance Formula
Understanding how to calculate distance is crucial for various real-life applications, including travel. With the distance formula, you can easily determine how far you've traveled based on your speed and the time spent traveling. The basic distance formula is given by:
  • Distance = Speed 脳 Time.
If you know the speed at which you're moving and the time elapsed, simply multiply these two values to find the distance.
For instance, in the case of Ellie's trip, her speed was 65 miles per hour and she traveled for 2.5 hours, making her total distance 65 脳 2.5 = 162.5 miles.
This demonstrates the straight-forward nature of the distance formula and how it can be applied to find distances for various trips.
Speed Calculation
Calculating speed requires understanding that it is a measure of how fast an object moves. To find the speed given distance and time, use the speed formula:
  • Speed = Distance 梅 Time.
This formula is particularly useful when you need to determine how quickly a journey was made.
When applying this to Ellie's grandmother's trip, you know the distance (162.5 miles) and the time taken (6 hours). Using the speed formula gives you the grandmother's speed: 162.5 梅 6 = 27.08 miles per hour.
This showcases how the formula can help in retrospectively understanding the travel conditions based on past trips.
Travel Time Comparison
Comparing travel times can highlight differences in travel conditions or advancements over time. This comparison can often reflect inverse variation principles, where two variables are related in such a way that as one increases, the other decreases.
In the context of Ellie and her grandmother's trips, both traveled the same distance. When Ellie traveled faster, her travel time was shorter at 2.5 hours. Meanwhile, her grandmother traveled slower, resulting in a longer travel time of 6 hours.
  • This scenario illustrates inverse variation, where the product of speed and time remains constant for the same distance.
  • As the speed went down (27.08 mph for her grandmother), the time went up (6 hours), maintaining the total distance of 162.5 miles.
Understanding this inverse relationship helps emphasize how changes in speed directly affect travel time, keeping the overall travel experience balanced.

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Most popular questions from this chapter

Mini-Investigation Consider the system $$ \left\\{\begin{array}{l} 3 x+2 y=7 \\ 2 x-y=4 \end{array}\right. $$ a. Solve each equation for \(y\) and graph the result on your calculator. Sketch the graph on your paper. b. Add the two original equations and solve the resulting equation for \(y\). Add this graph to your graph from \(12 \mathrm{a}\). What do you notice? c. Multiply the second original equation by 2 , then add this to the first equation. Solve this equation for \(x\) and add its graph to your graph from \(12 \mathrm{a}\). What do you notice? d. Multiply the first original equation by 2 and the second by \(-3\), then add the results. Solve this equation for \(y\) and add its graph to \(12 \mathrm{a}\). What do you notice? e. What is the solution to the system of equations? How does this point relate to the graphs you drew in \(12 \mathrm{a}-\mathrm{d}\) ? f. Write a few sentences summarizing any conjectures you can make based on this exercise.

Solve each equation for \(y\). a. \(3 x+4 y=5.2\) b. \(3(y-5)=2 x\)

Consider the equation \(5 x+2 y=10\). a. Solve the equation for \(y\) and sketch the graph. (ii) b. Multiply the equation \(5 x+2 y=10\) by 3 , and then solve for \(y\). How does the graph of this equation compare with the graph of the original equation? Explain your answer. (a)

Check that each ordered pair is a solution to each system. If the pair is not a solution point, explain why not. (71) a. \((-2,34)\) b. \((4.25,19.25)\) \(\left\\{\begin{array}{l}y=38+2 x \\ y=-21-0.5 x\end{array}\right.\) \(\left\\{\begin{array}{l}y=32-3 x \\ y=15+x\end{array}\right.\) c. \((2,12.3)\) $$ \left\\{\begin{array}{l} y=2.3+3.2 x \\ y=5.9+3.2 x \end{array}\right. $$

Solve the system of equations using the substitution method, and check your solution. (hi) $$ \left\\{\begin{array}{l} y=25+30 x \\ y=15+32 x \end{array}\right. $$
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