MTH120 College Algebra Chapter 2.3

 2.3 Models and Applications:

To set up a linear equation to solve a real-world application, you need to identify the relevant quantities, define your variables, and establish a relationship between those variables based on the problem statement. Here are the general steps to do so:

1. Read and Understand the Problem:

   • Carefully read the problem statement to understand the context and what is being asked.

2. Identify the Relevant Quantities:

   • Identify the quantities or variables involved in the problem. These are typically the values that change or are unknown.

3. Define Your Variables:

   • Choose letters or symbols to represent the unknown quantities as variables. Typically, use a single letter like $�$ or $�$ for each variable.

4. Establish Relationships:

   • Determine how the variables are related to each other based on the problem statement. This relationship is usually expressed as an equation.

5. Write the Equation:

   • Write the linear equation using your variables, constants, and mathematical operations (addition, subtraction, multiplication, division) to represent the relationship between the quantities.

6. Check Units and Consistency:

   • Ensure that the units of measurement for all quantities are consistent in the equation.

7. Solve the Equation:

   • Once the equation is set up, you can solve it to find the value of the unknown variable, which is often the solution to the real-world problem.

Let's work through an example:

Example: Suppose you are given a problem about a car rental service: "A car rental company charges a base fee of $30 plus $0.25 per mile driven. You rent a car, and the total cost is $55. How many miles did you drive?"

Step 1: Understand the Problem:

• The problem is about finding the number of miles driven based on the total cost of renting a car.

Step 2: Identify the Relevant Quantities:

• Relevant quantities include the total cost, the base fee, the cost per mile, and the number of miles driven.

Step 3: Define Your Variables:

• Let $�$ represent the total cost in dollars.
• Let $�$ represent the base fee in dollars.
• Let $�$ represent the number of miles driven.
• The cost per mile is given as $0.25 per mile, so we don't need a separate variable for it.

Step 4: Establish Relationships:

• The total cost ($�$) is equal to the base fee ($�$) plus the cost per mile times the number of miles driven ($0.25�$): $�=�+0.25�$

Step 5: Write the Equation:

• Write the equation based on the established relationship: $�=30+0.25�$

Step 6: Check Units and Consistency:

• The units are consistent (dollars for costs and miles for distance).

Step 7: Solve the Equation:

• Given that the total cost ($�$) is $55, you can solve for $�$: $55=30+0.25�$

• Subtract 30 from both sides: $0.25�=55-30$ $0.25�=25$

• Divide both sides by 0.25: $�=25\mathrm{/}0.25$ $�=100$

So, you drove 100 miles.

Let's work through an example of modeling a linear equation to fit a real-world problem.

Problem Statement: Suppose you are managing a lemonade stand, and you want to determine how much revenue you'll make based on the number of glasses of lemonade you sell. You know that you sell each glass for $1, and you also receive a fixed donation of $10 from a local supporter regardless of how many glasses you sell. How can you model your revenue (in dollars) as a function of the number of glasses of lemonade you sell?

Step 1: Understand the Problem:

• The problem is about modeling the revenue of a lemonade stand based on the number of glasses of lemonade sold.

Step 2: Identify the Relevant Quantities:

• Relevant quantities include revenue, the number of glasses of lemonade sold, and the fixed donation.

Step 3: Define Your Variables:

• Let $�$ represent the total revenue in dollars.
• Let $�$ represent the number of glasses of lemonade sold.
• The fixed donation is $10.

Step 4: Establish Relationships:

• The total revenue ($�$) is equal to the revenue from selling glasses of lemonade ($1�$) plus the fixed donation ($10$): $�=1�+10$

Step 5: Write the Equation:

• Write the linear equation based on the established relationship: $�=�+10$

Step 6: Check Units and Consistency:

• The units are consistent (dollars for revenue and glasses for quantity).

Step 7: Model the Equation:

• The equation $�=�+10$ models the revenue of the lemonade stand based on the number of glasses of lemonade sold. It indicates that for each glass sold, you earn $1, and you also receive a fixed donation of $10.

This equation can be used to predict your revenue for any number of glasses of lemonade sold. For example, if you sell 50 glasses, you can calculate your revenue as follows: $�=50+10=60$ So, your revenue would be $60.

You can also graph this equation to visualize the relationship between revenue and the number of glasses sold. The slope of the line is 1, indicating that for each additional glass sold, revenue increases by $1. The $10 donation is represented by the y-intercept, where revenue starts when no glasses are sold.

Let's work through an example of modeling a linear equation to solve an unknown number problem.

Problem Statement: Suppose you are given a problem related to ages. You know that the sum of the ages of two people is 30 years, and the older person is 8 years older than the younger person. How can you model this situation with a linear equation and find the ages of the two individuals?

Step 1: Understand the Problem:

• The problem is about finding the ages of two people based on the given information about their sum and the age difference.

Step 2: Identify the Relevant Quantities:

• Relevant quantities include the ages of the two people.

Step 3: Define Your Variables:

• Let $�$ represent the age of the older person.
• Let $�$ represent the age of the younger person.

Step 4: Establish Relationships:

• We know that the sum of their ages is 30 years: $�+�=30$.
• We also know that the older person is 8 years older than the younger person: $�=�+8$.

Step 5: Write the Equations:

• Write the two linear equations based on the established relationships:
  1. $�+�=30$
  2. $�=�+8$

Step 6: Solve the System of Equations:

• Now, you have a system of two equations with two variables:

  1. $�+�=30$
  2. $�=�+8$

  You can use various methods to solve this system of equations, such as substitution or elimination. Let's use substitution:

  From equation (2), isolate $�$: $�=�+8$

  Substitute this expression for $�$ into equation (1): $(�+8)+�=30$

  Simplify the equation: $2�+8=30$

  Subtract 8 from both sides: $2�=22$

  Divide by 2: $�=11$

  Now that you've found the value of $�$, you can find $�$ using equation (2): $�=�+8=11+8=19$

Step 7: Interpret the Results:

• The older person is 19 years old, and the younger person is 11 years old.

So, you've modeled this problem with a system of linear equations and solved for the ages of the two individuals. The older person is 19 years old, and the younger person is 11 years old.

To set up a linear equation to solve a real-world application, you'll need to follow these general steps:

Step 1: Understand the Problem:

• Carefully read and understand the problem statement to grasp the context and what information is provided.

Step 2: Identify the Relevant Quantities:

• Determine which quantities are involved in the problem and which ones are unknown or need to be determined.

Step 3: Define Your Variables:

• Choose meaningful variables to represent the unknown quantities. Typically, use letters like $�$, $�$, $�$, etc., for variables.

Step 4: Establish Relationships:

• Determine how the variables are related to each other based on the information provided in the problem.

Step 5: Write the Linear Equation:

• Use the relationships established in step 4 to write a linear equation in terms of the chosen variables. The equation should represent the connection between the quantities.

Step 6: Check Units and Consistency:

• Ensure that the units of measurement for all quantities are consistent in the equation.

Step 7: Interpret the Equation:

• Understand the meaning of the equation in the context of the problem. It should provide a way to calculate the unknown quantity based on known information.

Let's work through an example:

Example: Suppose you're planning a road trip and want to calculate the total cost of gasoline based on the number of miles driven and the price of gasoline. You know that your car's fuel efficiency is 25 miles per gallon (mpg), and the price of gasoline is $3.00 per gallon. How can you set up a linear equation to calculate the total cost of gasoline?

Step 1: Understand the Problem:

• The problem involves calculating the total cost of gasoline based on the number of miles driven and the price of gasoline.

Step 2: Identify the Relevant Quantities:

• Relevant quantities include the total cost, the number of miles driven, the fuel efficiency (mpg), and the price of gasoline.

Step 3: Define Your Variables:

• Let $�$ represent the total cost in dollars.
• Let $�$ represent the number of miles driven.
• Let $�$ represent the fuel efficiency in miles per gallon (25 mpg).
• Let $�$ represent the price of gasoline per gallon ($3.00).

Step 4: Establish Relationships:

• The total cost ($�$) is related to the number of miles driven ($�$), fuel efficiency ($�$), and the price of gasoline ($�$) by the formula: Total Cost = $(�\mathrm{/}�)\times �$.

Step 5: Write the Linear Equation:

• Write the linear equation based on the established relationship: $�=(�\mathrm{/}�)\times �$

Step 6: Check Units and Consistency:

• Ensure that the units are consistent. In this case, miles (M) and miles per gallon (E) are compatible.

Step 7: Interpret the Equation:

• The equation $�=(�\mathrm{/}�)\times �$ allows you to calculate the total cost of gasoline ($�$) based on the number of miles driven ($�$), the fuel efficiency ($�$), and the price of gasoline ($�$).

This equation provides a practical way to estimate your gasoline expenses for the road trip. Simply plug in the values of $�$, $�$, and $�$ to calculate the total cost.

To solve a real-world application using a formula, you'll need to follow these steps:

Step 1: Understand the Problem:

• Carefully read and understand the problem statement to grasp the context and what information is provided.

Step 2: Identify the Relevant Quantities:

• Determine which quantities are involved in the problem and which ones are unknown or need to be determined.

Step 3: Identify the Appropriate Formula:

• Based on the problem's context and the quantities involved, identify the formula or mathematical relationship that connects these quantities. If you're unsure, consult relevant mathematical concepts or formulas.

Step 4: Define Your Variables:

• Choose meaningful variables to represent the unknown quantities and any known values from the problem. Typically, use letters like $�$, $�$, $�$, etc., for variables.

Step 5: Plug in Known Values:

• Substitute the known values from the problem into the formula.

Step 6: Solve for the Unknown Variable:

• Use the formula, along with the known values, to calculate the unknown variable. If necessary, perform any algebraic manipulations to isolate the unknown variable.

Step 7: Interpret the Results:

• Understand the meaning of the calculated result in the context of the problem. Consider units of measurement and any relevant implications.

Let's work through an example:

Example: Suppose you're planning a trip to a city that's 300 miles away, and you want to calculate the travel time in hours. You know that your car's average speed is 60 miles per hour (mph). How can you use the formula $\text{time}=\frac{\text{distance}}{\text{speed}}$ to calculate the travel time?

Step 1: Understand the Problem:

• The problem involves calculating the travel time based on the distance to be traveled and the average speed of the car.

Step 2: Identify the Relevant Quantities:

• Relevant quantities include the travel time, the distance (300 miles), and the average speed (60 mph).

Step 3: Identify the Appropriate Formula:

• The appropriate formula for this problem is $\text{time}=\frac{\text{distance}}{\text{speed}}$.

Step 4: Define Your Variables:

• Let $�$ represent the travel time in hours.
• Let $�$ represent the distance in miles (300 miles).
• Let $�$ represent the average speed in miles per hour (60 mph).

Step 5: Plug in Known Values:

• Substitute the known values into the formula: $�=\frac{�}{�}=\frac{300\text{ miles}}{60\text{ mph}}$

Step 6: Solve for the Unknown Variable:

• Calculate the travel time: $�=\frac{300\text{ miles}}{60\text{ mph}}=5\text{ hours}$

Step 7: Interpret the Results:

• The travel time for the 300-mile trip at an average speed of 60 mph is 5 hours.

So, you've successfully used the formula to calculate the travel time for your trip.

To solve an application using a formula, you can follow these steps:

Step 1: Understand the Problem:

• Carefully read and understand the problem statement to grasp the context and what information is provided.

Step 2: Identify the Relevant Quantities:

• Determine which quantities are involved in the problem and which ones are unknown or need to be determined.

Step 3: Identify the Appropriate Formula:

• Based on the problem's context and the quantities involved, identify the formula or mathematical relationship that connects these quantities. This may require knowledge of specific mathematical concepts or equations.

Step 4: Define Your Variables:

• Choose meaningful variables to represent the unknown quantities and any known values from the problem. Typically, use letters like $�$, $�$, $�$, etc., for variables.

Step 5: Plug in Known Values:

• Substitute the known values from the problem into the formula.

Step 6: Solve for the Unknown Variable:

• Use the formula, along with the known values, to calculate the unknown variable. If necessary, perform any algebraic manipulations to isolate the unknown variable.

Step 7: Interpret the Results:

• Understand the meaning of the calculated result in the context of the problem. Consider units of measurement and any relevant implications.

Let's work through an example:

Example: Suppose you have a tank that contains 40 liters of water, and water is being drained from the tank at a rate of 2 liters per minute. You want to calculate how many minutes it will take for the tank to be completely empty.

Step 1: Understand the Problem:

• The problem involves calculating the time it takes for a tank to empty when water is being drained from it at a constant rate.

Step 2: Identify the Relevant Quantities:

• Relevant quantities include the initial amount of water (40 liters), the rate of drainage (2 liters per minute), and the time it takes for the tank to empty (unknown).

Step 3: Identify the Appropriate Formula:

• In this case, the appropriate formula is $\text{time}=\frac{\text{amount}}{\text{rate}}$.

Step 4: Define Your Variables:

• Let $�$ represent the time in minutes.
• Let $�$ represent the amount of water in the tank (in liters) at any given time.
• Let $�$ represent the drainage rate (2 liters per minute).

Step 5: Plug in Known Values:

• Substitute the known values into the formula: $�=\frac{�}{�}=\frac{40\text{ liters}}{2\text{ liters per minute}}$

Step 6: Solve for the Unknown Variable:

• Calculate the time it takes for the tank to empty: $�=\frac{40\text{ liters}}{2\text{ liters per minute}}=20\text{ minutes}$

Step 7: Interpret the Results:

• It will take 20 minutes for the tank to be completely empty when water is being drained from it at a rate of 2 liters per minute.

So, you've used the formula to calculate the time it takes for the tank to empty in this real-world application.

Solving a perimeter problem typically involves finding the total length of the boundary or outer edge of a geometric shape. Here's a general approach for solving such problems:

Step 1: Understand the Problem:

• Carefully read and understand the problem statement to determine the type of shape or object for which you need to find the perimeter.

Step 2: Identify the Relevant Quantities:

• Determine which lengths, sides, or dimensions are relevant to calculating the perimeter of the shape.

Step 3: Identify the Perimeter Formula:

• Depending on the shape (e.g., rectangle, triangle, circle), identify the appropriate formula for calculating the perimeter. Each shape has a specific formula to find its perimeter.

Step 4: Define Your Variables:

• Assign variables to represent the lengths or dimensions involved in calculating the perimeter. Use meaningful letters like $�$, $�$, $�$, etc.

Step 5: Plug in Known Values:

• Substitute the known values (the given dimensions or lengths) into the perimeter formula.

Step 6: Calculate the Perimeter:

• Perform the necessary mathematical operations to calculate the perimeter. This might involve addition, multiplication, or other operations specified by the formula.

Step 7: Interpret the Results:

• Understand the meaning of the calculated perimeter in the context of the problem. Consider units of measurement and any relevant implications.

Let's work through an example:

Example: Suppose you have a rectangular garden with a length of 10 feet and a width of 6 feet. Find the perimeter of the garden.

Step 1: Understand the Problem:

• The problem involves finding the perimeter of a rectangular garden based on its given dimensions.

Step 2: Identify the Relevant Quantities:

• Relevant quantities include the length and width of the rectangular garden.

Step 3: Identify the Perimeter Formula:

• For a rectangle, the formula to calculate the perimeter ($�$) is $�=2\times (\text{length}+\text{width})$.

Step 4: Define Your Variables:

• Let $�$ represent the perimeter.
• Let $�$ represent the length of the garden (10 feet).
• Let $�$ represent the width of the garden (6 feet).

Step 5: Plug in Known Values:

• Substitute the known values into the perimeter formula: $�=2\times (�+�)=2\times (10\text{ feet}+6\text{ feet})$

Step 6: Calculate the Perimeter:

• Perform the mathematical operations to calculate the perimeter: $�=2\times (10\text{ feet}+6\text{ feet})=2\times 16\text{ feet}=32\text{ feet}$

Step 7: Interpret the Results:

• The perimeter of the rectangular garden is 32 feet. This means that the total length of the boundary or outer edge of the garden is 32 feet.

So, you've successfully calculated the perimeter of the rectangular garden using the appropriate formula for a rectangle.

Solving an area problem involves finding the amount of space enclosed by a geometric shape or region. Here's a general approach for solving such problems:

Step 1: Understand the Problem:

• Carefully read and understand the problem statement to determine the type of shape or region for which you need to find the area.

Step 2: Identify the Relevant Quantities:

• Determine which dimensions, lengths, or side lengths are relevant to calculating the area of the shape or region.

Step 3: Identify the Area Formula:

• Depending on the shape (e.g., rectangle, triangle, circle), identify the appropriate formula for calculating the area. Each shape has a specific formula to find its area.

Step 4: Define Your Variables:

• Assign variables to represent the relevant dimensions or lengths involved in calculating the area. Use meaningful letters like $�$, $�$, $�$, $�$, etc.

Step 5: Plug in Known Values:

• Substitute the known values (the given dimensions or lengths) into the area formula.

Step 6: Calculate the Area:

• Perform the necessary mathematical operations to calculate the area. This might involve multiplication, division, or other operations specified by the formula.

Step 7: Interpret the Results:

• Understand the meaning of the calculated area in the context of the problem. Consider units of measurement and any relevant implications.

Let's work through an example:

Example: Suppose you have a rectangular room with a length of 12 feet and a width of 8 feet. Find the area of the room.

Step 1: Understand the Problem:

• The problem involves finding the area of a rectangular room based on its given dimensions.

Step 2: Identify the Relevant Quantities:

• Relevant quantities include the length and width of the rectangular room.

Step 3: Identify the Area Formula:

• For a rectangle, the formula to calculate the area ($�$) is $�=\text{length}\times \text{width}$.

Step 4: Define Your Variables:

• Let $�$ represent the area.
• Let $�$ represent the length of the room (12 feet).
• Let $�$ represent the width of the room (8 feet).

Step 5: Plug in Known Values:

• Substitute the known values into the area formula: $�=�\times �=12\text{ feet}\times 8\text{ feet}$

Step 6: Calculate the Area:

• Perform the mathematical operations to calculate the area: $�=12\text{ feet}\times 8\text{ feet}=96\text{ square feet}$

Step 7: Interpret the Results:

• The area of the rectangular room is 96 square feet. This represents the total amount of space enclosed by the room.

So, you've successfully calculated the area of the rectangular room using the appropriate formula for a rectangle.

Solving a volume problem involves finding the amount of space enclosed by a three-dimensional object or container. Here's a general approach for solving such problems:

Step 1: Understand the Problem:

• Carefully read and understand the problem statement to determine the type of three-dimensional object or container for which you need to find the volume.

Step 2: Identify the Relevant Quantities:

• Determine which dimensions, lengths, or side lengths are relevant to calculating the volume of the object or container.

Step 3: Identify the Volume Formula:

• Depending on the shape (e.g., cube, cylinder, sphere), identify the appropriate formula for calculating the volume. Each shape has a specific formula to find its volume.

Step 4: Define Your Variables:

• Assign variables to represent the relevant dimensions or lengths involved in calculating the volume. Use meaningful letters like $�$, $�$, $�$, $�$, $�$, etc.

Step 5: Plug in Known Values:

• Substitute the known values (the given dimensions or lengths) into the volume formula.

Step 6: Calculate the Volume:

• Perform the necessary mathematical operations to calculate the volume. This might involve multiplication, exponentiation, or other operations specified by the formula.

Step 7: Interpret the Results:

• Understand the meaning of the calculated volume in the context of the problem. Consider units of measurement and any relevant implications.

Let's work through an example:

Example: Suppose you have a cylindrical tank with a radius of 4 feet and a height of 10 feet. Find the volume of the tank.

Step 1: Understand the Problem:

• The problem involves finding the volume of a cylindrical tank based on its given dimensions.

Step 2: Identify the Relevant Quantities:

• Relevant quantities include the radius and height of the cylindrical tank.

Step 3: Identify the Volume Formula:

• For a cylinder, the formula to calculate the volume ($�$) is $�=�\times {\text{radius}}^2\times \text{height}$.

Step 4: Define Your Variables:

• Let $�$ represent the volume.
• Let $�$ represent the radius of the tank (4 feet).
• Let $ℎ$ represent the height of the tank (10 feet).

Step 5: Plug in Known Values:

• Substitute the known values into the volume formula: $�=�\times (4\text{ feet}{)}^2\times (10\text{ feet})$

Step 6: Calculate the Volume:

• Perform the mathematical operations to calculate the volume: $�=�\times 16\text{ square feet}\times 10\text{ feet}=160�\text{ cubic feet}$

Step 7: Interpret the Results:

• The volume of the cylindrical tank is $160�$ cubic feet. This represents the total amount of space enclosed by the tank.

So, you've successfully calculated the volume of the cylindrical tank using the appropriate formula for a cylinder.

Here are some real-world examples of volume problems and how to solve them:

1. Water Tank Volume:

• Problem: You have a cylindrical water tank with a radius of 3 meters and a height of 5 meters. Calculate the volume of water the tank can hold.
• Solution:
  • Identify the relevant quantities: radius ($�$) and height ($ℎ$).
  • Use the volume formula for a cylinder: $�=�\times {�}^2\times ℎ$.
  • Plug in the values: $�=�\times (3\text{m}{)}^2\times 5\text{m}=45�\text{cubic meters}$.
  • Interpretation: The tank can hold 45π cubic meters of water.

2. Shipping Container Volume:

• Problem: You have a rectangular shipping container with dimensions 8 feet by 6 feet by 4 feet. Find the volume of the container.
• Solution:
  • Identify the relevant quantities: length ($�$), width ($�$), and height ($�$).
  • Use the volume formula for a rectangular box: $�=�\times �\times �$.
  • Plug in the values: $�=8\text{ft}\times 6\text{ft}\times 4\text{ft}=192\text{cubic feet}$.
  • Interpretation: The container has a volume of 192 cubic feet.

3. Swimming Pool Volume:

• Problem: You have an oval-shaped swimming pool with a length of 20 meters and a width of 10 meters. The depth of the pool varies from 2 meters at one end to 4 meters at the other end. Calculate the volume of the pool.
• Solution:
  • Identify the relevant quantities: length ($�$), width ($�$), and varying depth.
  • Use the formula for the volume of an oval-shaped pool: $�=\frac{1}{3}\times �\times �\times �\times (\text{average depth})$.
  • Calculate the average depth: $\text{Average depth}=\frac{\text{depth at one end}+\text{depth at the other end}}{2}=\frac{2\text{m}+4\text{m}}{2}=3\text{m}$.
  • Plug in the values: $�=\frac{1}{3}\times �\times 20\text{m}\times 10\text{m}\times 3\text{m}=600�\text{cubic meters}$.
  • Interpretation: The swimming pool has a volume of 600π cubic meters.

These are practical examples of volume problems that can be encountered in various real-world scenarios, from water tanks to shipping containers to swimming pools. The key is to identify the appropriate formula for the specific shape and then plug in the relevant dimensions to calculate the volume.