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Two or more quantities can have a constant ratio. For example, the circumference of any circle is proportional to its diameter, and its ratio is always equal to a constant, In this lesson, equations in two or more variables will be created to represent relationships between quantities and their graphs will be analyzed.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

## Number of Phone Calls Per Day Between Two Cities

The average number of phone calls per day between two cities varies directly with the populations of the cities and and inversely with the square of distance between the two cities.

External credits: TUBS
a Which equation models the given situation?
b In the population of San Francisco was about and the population of Portland was about The distance between the two cities is about miles. In the given year, the average number of calls between the cities was about Use the model determined in Part A to Find the constant Round the answer to three decimal places.
c In the average number of daily phone calls between San Francisco and Los Angeles, with a population of about was about Use the model to find the approximate distance between the two cities.
Discussion

## Joint Variation

A joint variation, also known as joint proportionality, occurs when one variable varies directly with two or more variables. In other words, if a variable varies directly with the product of other variables, it is called joint variation.

Here, the variable varies jointly with and and is the constant of variation. Here are some examples of joint variation.

Examples of Joint Variation
Example Rule Comment
The area of a rectangle Here, is the rectangle's length, its width, and the constant of variation is
The volume of a pyramid Here, and are the length and the width of the base, respectively, while is the pyramid's height. The constant of variation is
Lastly, it is important to note that joint variation is closely related to other types of variation.
Example

## Television Series

Vincenzo and Emily are having a lively chat about television series they love. Emily managed to watch episodes of The Flash in just days! Each episode typically lasts minutes.

Emily and Vincenzo agree that the number of days it takes to watch an entire show is jointly proportional to the number of episodes and the length of the episodes
Vincenzo wants to watch Sherlock, which has episodes, and each episode is about minutes long. How long will it take to watch the series if Vincenzo watch the series at the same rate as Emily did? Round the answer to the nearest integer.

### Hint

Use the fact that if varies jointly with and the equation of variation is , where is the constant of variation.

### Solution

Begin with recalling that if varies jointly with and the equation of variation is
Here, is the constant of variation and In this case, the number of days it takes to watch a show is jointly proportional to the number of episodes and the length of the episodes This means the equation of the variation can be written as follows.
Now, will be determined by using the given information about the series Emily watched. It is given that it took days to watch episodes that are each about minutes long. Substitute and into the equation and solve for
Solve for
Using the time it will take to watch Sherlock can be determined. Recall that it is given that there are episodes and each is minutes long.
Evaluate right-hand side
It will take Vincenzo about days to watch Sherlock. Time to stock up on snacks!
Explore

## Recognzing Inverse Variation

Use the applet to investigate the relationship between the width and the length of rectangles.
For rectangles with a fixed area, say square units, complete the table.
Width Length
Draw a scatter plot of the data. Does it make sense to say that the width varies inversely with the length? Explain.
Discussion

## Inverse Variation

An inverse variation, or inverse proportionality, occurs when two non-zero variables have a relationship such that their product is constant. This relationship is often written with one of the variables isolated on the left-hand side.

The constant is the constant of variation. When the relationship is not an inverse variation. In the following example, the constant of variation is
The constant of variation may be any real number except Here are some examples of inverse variations.
Examples of Inverse Variation
Example Rule Comment
The gas pressure in a sealed container if the container's volume is changed, given constant temperature and constant amount of gas. The variables are the pressure and the volume The amount of gas temperature and universal gas constant are fixed values. Therefore, the constant of variation is
The time it takes to travel a given distance at various speeds. The constant of variation is the distance and the variables are the time and the speed
Inverse variation is closely related to other types of variation.
Pop Quiz

## Identifying the Type of Variation

Determine whether the relationship between the variables in the table shows direct or inverse variation, or neither.

Example

## Number of Songs on Emily's Phone

Emily, tired of watching shows, wants to update the playlist on her phone before starting a family road trip from Portland to San Francisco. The number of songs that can be stored on her phone varies inversely with the average size of a song.

Emily's phone can store songs when the average size of a song is megabytes (MB).

a How many songs with an average size of megabytes does Emily's phone store?
b List the number of songs that will fit on the phone when the average size of a song is and respectively. Round the answer to the nearest integer whenever necessary.

### Hint

a Begin by finding the constant of variation.
b Use the inverse variation equation to create a table.

### Solution

a It is given that the number of songs that can be stored on Emily's phone varies inversely with the average size of a song. Recall the form of an inverse variation equation that relates and
In the equation, is the constant of variation. It is known that when the average size of a song is megabytes, the phone can store up to songs. Therefore, substitute and into the inverse variation equation and solve for
The constant of variation is Using this information, the number of songs with an average size of can be found.
b In the previous part, the inverse variation equation was found.
Now a table that shows the number of songs when the average size of a song is and can be made.
Size, Number of Songs,

In the table, as the size gets larger, the number of songs that the phone can store gets smaller. Therefore, the number of songs decreases as the average size increases.

Example

## Emily's Trip to San Francisco

Now that the updated playlist and everything else is ready, Emily's journey from Portland to San Francisco can begin.
External credits: TUBS
The time it takes to reach San Francisco varies inversely with their average rate of speed.
a If Emily lives just outside of the main City of Portland miles from San Francisco by car, write an equation that relates the travel time to the average speed. In addition to the equation, draw its corresponding graph.
b Determine the minimum average speed that Emily would need to drive to reach San Francisco within hours. Round the answer to the nearest integer.

a Example Equation:

Example Graph:

### Hint

a Use the standard equation of the inverse variation, where is the constant of variation.
b Substitute for the time into the equation from Part A and solve for the rate of speed.

### Solution

a It is given that the time it takes Emily to reach San Francisco varies inversely with her average rate of speed. Let be the time and be the rate of speed. Then, the following equation can be written.
Here, is a constant of variation. It is also given that Emily lives miles away from San Francisco. This is the value of Recall that a distance is a product of the time and rate of speed. Thus, the time can be expressed as a quotient of the distance by the rate of speed.
Comparing this formula with the equation, it is seen that indeed represents the distance. Therefore, can be substituted with
Now make a table of values to graph the equation.

Ordered pairs are the coordinates of the points on the graph. Plot the points and connect them with a smooth curve.

b To determine the minimum average speed that will allow Emily to arrive to San Francisco within hours, substitute for into the equation from Part A and solve it for
Emily should travel at a minimum speed of miles per hour.
Discussion

## Combined Variation

A combined variation, or combined proportionality, occurs when one variable depends on two or more variables, either directly, inversely, or a combination of both. This means that any joint variation is also a combined variation.

The variable varies directly with and inversely with and is the constant of variation. Therefore, this is a combined variation. Here are some examples.

Examples of Combined Variation
Example Rule Comment
Newton's Law of Gravitational Force The gravitational force varies directly as the masses of the objects and and inversely as the square of the distance between the objects. The gravitational constant is the constant of variation.
The Ideal Gas Law The pressure varies directly as the number of moles and the temperature and inversely as the volume The universal gas constant is the constant of variation.
Combined variation is closely related to other types of variation.
Example

## Number of T-Shirts Sold

Emily is wandering around a gift shop to buy gifts for some of her friends. Emily overhears a conversation between the shopkeeper and an employee. The shopkeeper says that the number of t-shirts sold is directly proportional to their advertising budget and inversely proportional to the price of each t-shirt.

When are spent on advertising and the price of each t-shirt is the number of t-shirts sold is How many t-shirts are sold when the advertising budget is and the price of each t-shirt is

### Hint

Use the equation of the combined variation, where is the constant of variation.

### Solution

When one quantity varies with respect to two or more quantities, this variation can be regarded as a combined variation.

Combined Variation Equation Form
varies jointly with and
varies jointly with and and inversely with
varies directly with and inversely with the product
Based on this table, an equation that models the given variation can be written. It is given that the number of t-shirts sold varies directly with the advertising budget and inversely with the price of a t-shirt
Here is the constant of variation and cannot be With an advertising budget of and the t-shirt price of t-shirts are sold. Using this information, the value of can be found. To do so, substitute and in the above equation.
Solve for
Now that the value of is known, the equation that models the variation can be written.
The shopkeeper increases the budget to and the price of a t-shirt to To find the value of the under these circumstances, substitute and into the equation.
Evaluate right-hand side

### Alternative Solution

The number of t-shirts sold varies directly with the advertising budget and inversely with the price of a t-shirt Additionally, when and The goal is to find when and To find the value of write two equations that involve the constant of variation
Since is equal to both and by the Transitive Property of Equality these two expressions must be equal. Using this information, a proportion can be written.
Next, substitute and and solve for
Solve for
Pop Quiz

## Finding the Value of

In the applet, various types of variations are shown randomly. Find the value of by using the given values. If necessary, round the answer to the two decimal places.

Closure

## Number of Phone Calls Per Day Between Two Cities

In this lesson, variation types are explained with real-life examples. Considering those examples, the challenge presented at the beginning of the lesson can be solved with confidence. Recall that the average number of phone calls per day between two cities varies directly with the populations of the cities and inversely with the square of the distance between the two cities.

External credits: TUBS
a Which equation models the given situation?
b In the population of San Francisco was about and the population of Portland was about The distance between the two cities is roughly miles. In the given year, the average number of calls between the cities was about Use the model determined in Part A to Find the constant Round the answer to three decimal places.
c In the average number of daily phone calls between San Francisco and Los Angeles, with a population of about was about Use the model to find the approximate distance between the two cities.

### Hint

a Use the equation of the combined variation, where is the constant of variation.
b Substitute the given values to find the value of
c Substitute the given values into the variation equation.

### Solution

a Recall the labels of the quantities.
It is known that varies directly with and and inversely with the square of Therefore, is equal to the product of and divided by the square of
Here is the constant of variation and cannot be This equation models the given variation.
b The value of can be found by using the given information and the equation written in Part A.
San Francisco Portland
Population
Distance
Number of Calls
These values represent and Now, substitute them into the equation.
Solve for
The constant of variation is about
c Using the value of found in Part B, the equation that models the variation can be expressed.
For clarity, make a table to organize the given information.
San Francisco Los Angeles
Population
Distance
Number of Calls
To find the value of the substitute and into the equation and solve for
Solve for

Since the distance cannot be negative, choose the positive value and not that the distance between the two cities is miles.