Problem 30 - Elementary Algebra

Question

When a car travels a fixed distance, the relationship between the speed of the car, \(x\), and the time it travels, \(y\), is an inverse variation. When the speed is \(\frac{48 \mathrm{mi}}{1 \mathrm{hr}}\), the time is \(0.75 \mathrm{hr}\). a. Find the constant of proportionality. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the time in hours to travel this distance at a speed of \(\frac{80 \mathrm{mi}}{1 \mathrm{hr}}\). d. Change the time in part \(\mathrm{c}\) to minutes.

Step-by-Step Solution

Answer

The constant of proportionality is 36 mi. The equation is \(y = \frac{36}{x}\). At 80 mi/hr, the time is 0.45 hours or 27 minutes.

Step 1: Identify the relationship

The problem states that the relationship between the car's speed, represented as \(x\), and the time to travel a fixed distance, represented as \(y\), is an inverse variation. This can be written as \(y = \frac{k}{x}\), where \(k\) is the constant of proportionality.

Step 2: Calculate the constant of proportionality

Use the given speed and time to find \(k\). When \(x = 48\) mi/hr and \(y = 0.75\) hr, substitute these values into the equation. \[0.75 = \frac{k}{48}\] Solve for \(k\): \(k = 0.75 \times 48 = 36 \) mi.

Step 3: Write the equation that represents the relationship

Using the calculated constant of proportionality, the equation becomes: \(y = \frac{36}{x}\).

Step 4: Find the time to travel at a speed of 80 mi/hr

Substitute \(x = 80\) mi/hr into the equation \(y = \frac{36}{x}\). \[y = \frac{36}{80} = 0.45 \text{ hours}\]

Step 5: Convert the time from hours to minutes

There are 60 minutes in 1 hour. Multiply the time found in Step 4 by 60 to convert it to minutes: \(0.45 \text{ hours} \times 60 = 27 \text{ minutes}\).

Key Concepts

constant of proportionality

In problems involving inverse variation, the idea of a constant of proportionality is crucial. The constant of proportionality, often denoted as \( k \), is a value that remains unchanged when the variables change. For inverse variation, the relationship between the two variables, speed (\( x \)) and time (\( y \)), can be described by \( y = \frac{k}{x} \).
To find \( k \), you need data points for both variables. For instance, if a car travels at 48 miles per hour, and the time taken is 0.75 hours, substitute these values into the equation to solve for \( k \).
\[ 0.75 = \frac{k}{48} \]
Rearrange to get:
\[ k = 0.75 \cdot 48 \]
This gives \( k = 36 \). The constant of proportionality here is 36 miles. This means that for any speed, the product of speed and time will always equal 36 miles.

speed and time relationship

The relationship between speed and time in an inverse variation can be intuitively understood. If you need to drive a fixed distance, increasing your speed decreases the time required, and decreasing your speed increases the time.
This relationship is mathematically expressed as \( y = \frac{k}{x} \), where \( k \) is the constant of proportionality, \( x \) is the speed, and \( y \) is the time. Given that \( k = 36 \) from our previous example, the equation becomes:
\[ y = \frac{36}{x} \]
This shows how time changes with speed. For instance, to find the time to travel the same distance at a different speed, like 80 miles per hour, substitute \ x = 80 \ into the equation:
\[ y = \frac{36}{80} = 0.45 \]
This means it will take 0.45 hours to cover the distance at 80 miles per hour. The inverse variation equation helps predict travel time for any given speed.

unit conversion

In mathematical problems and real-life situations, unit conversion is essential, especially when dealing with time and speed. Typically, we measure time in hours and minutes. Converting between these units requires understanding the relationships:
\ 1 \text{ hour } = 60 \text{ minutes} \
To convert from hours to minutes, multiply by 60. For example, if the travel time is 0.45 hours, convert it to minutes by:
\[ 0.45 \text{ hours} \times 60 = 27 \text{ minutes} \]
Such conversions are useful for accurate and user-friendly representations of time. Always ensure the correctness of units in your calculations to avoid errors and misinterpretations. Strong skills in unit conversion enable seamless transitions between different measurement systems.