Select The Correct Answer.Lance Is Knitting A Blanket And Needs To Buy Some More Yarn. At His Local Craft Store, 2 Balls Of Yarn Cost $$ 5$ And 8 Balls Of Yarn Cost $$ 20$[/tex]. What Is The Constant Of Proportionality In
What is Constant of Proportionality?
In mathematics, the constant of proportionality is a value that represents the ratio of two quantities that are directly proportional to each other. In other words, it is a constant value that describes the relationship between two variables that are directly proportional. The constant of proportionality is often denoted by the letter 'k' and is calculated by dividing the dependent variable by the independent variable.
Problem: Finding the Constant of Proportionality
Lance is knitting a blanket and needs to buy some more yarn. At his local craft store, 2 balls of yarn cost $5 and 8 balls of yarn cost $20. We need to find the constant of proportionality in this situation.
Step 1: Understand the Problem
We are given two pieces of information:
- 2 balls of yarn cost $5
- 8 balls of yarn cost $20
We need to find the constant of proportionality, which represents the ratio of the number of balls of yarn to the cost of the yarn.
Step 2: Identify the Variables
Let's identify the variables in this problem:
- Independent variable: number of balls of yarn (x)
- Dependent variable: cost of yarn (y)
Step 3: Write the Equation
We can write the equation of proportionality as:
y = kx
where k is the constant of proportionality.
Step 4: Find the Constant of Proportionality
We can use the given information to find the constant of proportionality. Let's use the first piece of information: 2 balls of yarn cost $5.
y = 5 x = 2
We can substitute these values into the equation:
5 = k(2)
To find the constant of proportionality, we can divide both sides of the equation by 2:
k = 5/2 k = 2.5
Step 5: Check the Answer
Let's check our answer by using the second piece of information: 8 balls of yarn cost $20.
y = 20 x = 8
We can substitute these values into the equation:
20 = k(8)
To find the constant of proportionality, we can divide both sides of the equation by 8:
k = 20/8 k = 2.5
Our answer checks out!
Conclusion
The constant of proportionality in this situation is 2.5. This means that for every 2 balls of yarn, the cost is $5, and for every 8 balls of yarn, the cost is $20.
Real-World Applications
The concept of constant of proportionality has many real-world applications, such as:
- Economics: understanding the relationship between the price of a product and the quantity demanded
- Physics: understanding the relationship between the force applied to an object and the distance it travels
- Engineering: understanding the relationship between the voltage applied to a circuit and the current flowing through it
Tips and Tricks
- When working with proportions, make sure to identify the independent and dependent variables.
- Use the equation of proportionality to find the constant of proportionality.
- Check your answer by using different values of the variables.
Practice Problems
- A bakery sells 2 dozen cupcakes for $15. What is the constant of proportionality between the number of cupcakes and the cost?
- A car travels 25 miles per gallon of gasoline. What is the constant of proportionality between the distance traveled and the amount of gasoline consumed?
Answer Key
- k = 7.5
- k = 25
Q&A: Constant of Proportionality =====================================
Frequently Asked Questions
Q: What is the constant of proportionality?
A: The constant of proportionality is a value that represents the ratio of two quantities that are directly proportional to each other. It is a constant value that describes the relationship between two variables that are directly proportional.
Q: How do I find the constant of proportionality?
A: To find the constant of proportionality, you need to identify the independent and dependent variables, write the equation of proportionality, and then solve for the constant of proportionality using the given information.
Q: What is the difference between direct and inverse proportionality?
A: Direct proportionality is a relationship between two variables where the ratio of the variables is constant. Inverse proportionality is a relationship between two variables where the product of the variables is constant.
Q: Can I use the constant of proportionality to make predictions?
A: Yes, you can use the constant of proportionality to make predictions about the relationship between the variables. For example, if you know the constant of proportionality and the value of one variable, you can use it to predict the value of the other variable.
Q: How do I check my answer for the constant of proportionality?
A: To check your answer, you can use different values of the variables and see if the constant of proportionality holds true. You can also use the equation of proportionality to check your answer.
Q: What are some real-world applications of the constant of proportionality?
A: The constant of proportionality has many real-world applications, such as:
- Economics: understanding the relationship between the price of a product and the quantity demanded
- Physics: understanding the relationship between the force applied to an object and the distance it travels
- Engineering: understanding the relationship between the voltage applied to a circuit and the current flowing through it
Q: Can I use the constant of proportionality to solve problems involving rates and ratios?
A: Yes, you can use the constant of proportionality to solve problems involving rates and ratios. For example, if you know the constant of proportionality and the rate at which something is happening, you can use it to find the ratio of the variables.
Q: How do I use the constant of proportionality to solve problems involving proportions?
A: To use the constant of proportionality to solve problems involving proportions, you need to identify the independent and dependent variables, write the equation of proportionality, and then solve for the constant of proportionality using the given information.
Q: Can I use the constant of proportionality to solve problems involving percentages?
A: Yes, you can use the constant of proportionality to solve problems involving percentages. For example, if you know the constant of proportionality and the percentage increase or decrease, you can use it to find the new value of the variable.
Q: How do I use the constant of proportionality to solve problems involving similar figures?
A: To use the constant of proportionality to solve problems involving similar figures, you need to identify the independent and dependent variables, write the equation of proportionality, and then solve for the constant of proportionality using the given information.
Q: Can I use the constant of proportionality to solve problems involving geometry?
A: Yes, you can use the constant of proportionality to solve problems involving geometry. For example, if you know the constant of proportionality and the dimensions of a shape, you can use it to find the area or perimeter of the shape.
Q: How do I use the constant of proportionality to solve problems involving trigonometry?
A: To use the constant of proportionality to solve problems involving trigonometry, you need to identify the independent and dependent variables, write the equation of proportionality, and then solve for the constant of proportionality using the given information.
Q: Can I use the constant of proportionality to solve problems involving calculus?
A: Yes, you can use the constant of proportionality to solve problems involving calculus. For example, if you know the constant of proportionality and the derivative of a function, you can use it to find the rate of change of the function.
Conclusion
The constant of proportionality is a powerful tool for solving problems involving proportions, rates, and ratios. By understanding the concept of constant of proportionality and how to use it to solve problems, you can become a more confident and proficient problem-solver.