The Cost, \[$ C \$\], Of A Ham Sandwich At A Deli Varies Directly With The Number Of Sandwiches, \[$ N \$\]. If \[$ C = \$54 \$\] When \[$ N \$\] Is 9, What Is The Cost Of The Sandwiches When \[$ N \$\] Is 3?A.

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Introduction

In this article, we will explore the concept of direct variation and how it can be applied to real-world problems. We will use the example of a ham sandwich at a deli to demonstrate how the cost of the sandwich varies directly with the number of sandwiches ordered.

What is Direct Variation?

Direct variation is a relationship between two variables where one variable is a constant multiple of the other variable. In other words, as one variable increases, the other variable also increases at a constant rate. This relationship can be represented mathematically as:

y = kx

where y is the dependent variable, x is the independent variable, and k is the constant of variation.

The Problem

In this problem, we are given that the cost of a ham sandwich at a deli varies directly with the number of sandwiches ordered. We are also given that the cost of the sandwich is $54 when 9 sandwiches are ordered. We need to find the cost of the sandwiches when 3 sandwiches are ordered.

Step 1: Write the Equation

Since the cost of the sandwich varies directly with the number of sandwiches ordered, we can write the equation as:

c = kn

where c is the cost of the sandwich, n is the number of sandwiches ordered, and k is the constant of variation.

Step 2: Find the Constant of Variation

We are given that the cost of the sandwich is $54 when 9 sandwiches are ordered. We can substitute these values into the equation to find the constant of variation:

54 = k(9)

To solve for k, we can divide both sides of the equation by 9:

k = 54/9

k = 6

So, the constant of variation is 6.

Step 3: Write the Equation with the Constant of Variation

Now that we have found the constant of variation, we can write the equation as:

c = 6n

Step 4: Find the Cost of the Sandwiches when 3 Sandwiches are Ordered

We need to find the cost of the sandwiches when 3 sandwiches are ordered. We can substitute n = 3 into the equation:

c = 6(3)

c = 18

So, the cost of the sandwiches when 3 sandwiches are ordered is $18.

Conclusion

In this article, we have demonstrated how to use direct variation to solve a real-world problem. We have shown how to write the equation, find the constant of variation, and use the equation to find the cost of the sandwiches when 3 sandwiches are ordered. This problem is a great example of how direct variation can be used to model real-world relationships.

Direct Variation Formula

The direct variation formula is:

y = kx

where y is the dependent variable, x is the independent variable, and k is the constant of variation.

Example Problems

  1. The cost of a book varies directly with the number of books ordered. If the cost of the book is $20 when 2 books are ordered, what is the cost of the book when 5 books are ordered?
  2. The distance traveled by a car varies directly with the time traveled. If the car travels 120 miles in 4 hours, what is the distance traveled by the car in 6 hours?

Solutions

  1. Let c be the cost of the book and n be the number of books ordered. We can write the equation as:

c = kn

We are given that the cost of the book is $20 when 2 books are ordered. We can substitute these values into the equation to find the constant of variation:

20 = k(2)

To solve for k, we can divide both sides of the equation by 2:

k = 20/2

k = 10

So, the constant of variation is 10.

Now that we have found the constant of variation, we can write the equation as:

c = 10n

We need to find the cost of the book when 5 books are ordered. We can substitute n = 5 into the equation:

c = 10(5)

c = 50

So, the cost of the book when 5 books are ordered is $50.

  1. Let d be the distance traveled by the car and t be the time traveled. We can write the equation as:

d = kt

We are given that the car travels 120 miles in 4 hours. We can substitute these values into the equation to find the constant of variation:

120 = k(4)

To solve for k, we can divide both sides of the equation by 4:

k = 120/4

k = 30

So, the constant of variation is 30.

Now that we have found the constant of variation, we can write the equation as:

d = 30t

We need to find the distance traveled by the car in 6 hours. We can substitute t = 6 into the equation:

d = 30(6)

d = 180

So, the distance traveled by the car in 6 hours is 180 miles.

Direct Variation in Real-World Applications

Direct variation is a fundamental concept in mathematics that has numerous real-world applications. Some examples include:

  • Physics: The distance traveled by an object varies directly with the time traveled.
  • Economics: The cost of a product varies directly with the number of products produced.
  • Biology: The growth rate of a population varies directly with the number of individuals in the population.

Q: What is direct variation?

A: Direct variation is a relationship between two variables where one variable is a constant multiple of the other variable. In other words, as one variable increases, the other variable also increases at a constant rate.

Q: How is direct variation represented mathematically?

A: Direct variation is represented mathematically as:

y = kx

where y is the dependent variable, x is the independent variable, and k is the constant of variation.

Q: What is the constant of variation?

A: The constant of variation is a number that represents the rate at which the dependent variable changes in response to changes in the independent variable.

Q: How do I find the constant of variation?

A: To find the constant of variation, you can use the following steps:

  1. Write the equation of direct variation.
  2. Substitute the values of the dependent and independent variables into the equation.
  3. Solve for the constant of variation.

Q: What are some examples of direct variation in real-world applications?

A: Some examples of direct variation in real-world applications include:

  • Physics: The distance traveled by an object varies directly with the time traveled.
  • Economics: The cost of a product varies directly with the number of products produced.
  • Biology: The growth rate of a population varies directly with the number of individuals in the population.

Q: How do I use direct variation to solve problems?

A: To use direct variation to solve problems, you can follow these steps:

  1. Write the equation of direct variation.
  2. Substitute the values of the dependent and independent variables into the equation.
  3. Solve for the unknown variable.

Q: What are some common mistakes to avoid when working with direct variation?

A: Some common mistakes to avoid when working with direct variation include:

  • Not writing the equation of direct variation correctly.
  • Not substituting the correct values into the equation.
  • Not solving for the unknown variable correctly.

Q: How do I determine if a relationship is a direct variation?

A: To determine if a relationship is a direct variation, you can use the following steps:

  1. Graph the relationship.
  2. Check if the graph is a straight line.
  3. If the graph is a straight line, then the relationship is a direct variation.

Q: What are some real-world examples of direct variation?

A: Some real-world examples of direct variation include:

  • The cost of a product varies directly with the number of products produced.
  • The distance traveled by a car varies directly with the time traveled.
  • The growth rate of a population varies directly with the number of individuals in the population.

Q: How do I use direct variation to model real-world relationships?

A: To use direct variation to model real-world relationships, you can follow these steps:

  1. Identify the dependent and independent variables.
  2. Write the equation of direct variation.
  3. Substitute the values of the dependent and independent variables into the equation.
  4. Solve for the unknown variable.

Q: What are some benefits of using direct variation to model real-world relationships?

A: Some benefits of using direct variation to model real-world relationships include:

  • Improved accuracy: Direct variation can be used to model complex relationships with high accuracy.
  • Simplified problem-solving: Direct variation can be used to simplify problem-solving by reducing the number of variables.
  • Increased understanding: Direct variation can be used to increase understanding of complex relationships by providing a clear and concise model.

Q: What are some limitations of using direct variation to model real-world relationships?

A: Some limitations of using direct variation to model real-world relationships include:

  • Assumes a linear relationship: Direct variation assumes a linear relationship between the dependent and independent variables.
  • Does not account for non-linear relationships: Direct variation does not account for non-linear relationships between the dependent and independent variables.
  • May not be applicable to all situations: Direct variation may not be applicable to all situations, such as relationships with multiple variables or non-linear relationships.