The Relationship Between Cup Capacity and Number of Cups Filled

Oct 13, 2025 by leolavanderia.com.br 74 views

The Relationship Between Cup Capacity and Number of Cups Filled

Understanding the Problem

The problem states that the number of cups of sports drink that can be filled from a cooler varies inversely with the capacity of each cup. This means that as the capacity of each cup increases, the number of cups that can be filled decreases, and vice versa. We are given that each cup holds 6 ounces and that 128 cups can be filled.

Defining the Inverse Variation

In an inverse variation, the product of the two variables is constant. Let's denote the number of cups as $c$ and the capacity of each cup as $x$ . We can write the equation for inverse variation as:

cx = k

where $k$ is the constant of variation.

Finding the Constant of Variation

We are given that each cup holds 6 ounces, so $x = 6$ . We are also given that 128 cups can be filled, so $c = 128$ . We can substitute these values into the equation to find the constant of variation:

128(6) = k

768 = k

Writing the Equation

Now that we have found the constant of variation, we can write the equation for the number of cups that can be filled as a function of the capacity of each cup:

c = \frac{k}{x}

c = \frac{768}{x}

Solving for the Number of Cups

We can use this equation to find the number of cups that can be filled for a given capacity of each cup. For example, if each cup holds 8 ounces, we can substitute $x = 8$ into the equation:

c = \frac{768}{8}

c = 96

This means that if each cup holds 8 ounces, 96 cups can be filled.

Example 2

If each cup holds 12 ounces, we can substitute $x = 12$ into the equation:

c = \frac{768}{12}

c = 64

This means that if each cup holds 12 ounces, 64 cups can be filled.

Conclusion

In this article, we have explored the relationship between the capacity of each cup and the number of cups that can be filled. We have defined the inverse variation and found the constant of variation. We have then written the equation for the number of cups that can be filled as a function of the capacity of each cup. Finally, we have solved for the number of cups that can be filled for different capacities of each cup.

Key Takeaways

The number of cups of sports drink that can be filled from a cooler varies inversely with the capacity of each cup.
The equation for inverse variation is $cx = k$ , where $k$ is the constant of variation.
The constant of variation is $k = 768$ .
The equation for the number of cups that can be filled as a function of the capacity of each cup is $c = \frac{k}{x}$ .
We can use this equation to find the number of cups that can be filled for a given capacity of each cup.

Further Exploration

This problem can be extended to explore other relationships between variables. For example, we could explore the relationship between the number of cups and the volume of the cooler. We could also explore the relationship between the capacity of each cup and the amount of sports drink that can be filled.

References

[1] "Inverse Variation". Math Open Reference. Retrieved 2023-02-20.
[2] "Inverse Variation Formula". Mathway. Retrieved 2023-02-20.

Glossary

Inverse variation: A relationship between two variables in which the product of the two variables is constant.
Constant of variation: A constant value that is used in an inverse variation equation.
Capacity: The amount of a substance that a container can hold.
Number of cups: The number of cups that can be filled with a substance.
The Relationship Between Cup Capacity and Number of Cups Filled: Q&A

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about the relationship between cup capacity and the number of cups that can be filled.

Q: What is the relationship between cup capacity and the number of cups that can be filled?

A: The number of cups of sports drink that can be filled from a cooler varies inversely with the capacity of each cup. This means that as the capacity of each cup increases, the number of cups that can be filled decreases, and vice versa.

Q: How do I calculate the number of cups that can be filled for a given capacity of each cup?

A: To calculate the number of cups that can be filled, you can use the equation:

c = \frac{k}{x}

where $c$ is the number of cups, $k$ is the constant of variation (which is 768 in this case), and $x$ is the capacity of each cup.

Q: What is the constant of variation?

A: The constant of variation is a constant value that is used in an inverse variation equation. In this case, the constant of variation is 768.

Q: How do I find the constant of variation?

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

cx = k

where $c$ is the number of cups, $x$ is the capacity of each cup, and $k$ is the constant of variation.

Q: What is the relationship between the capacity of each cup and the amount of sports drink that can be filled?

A: The capacity of each cup and the amount of sports drink that can be filled are related through the equation:

c = \frac{k}{x}

where $c$ is the number of cups, $k$ is the constant of variation, and $x$ is the capacity of each cup.

Q: Can I use this equation to find the number of cups that can be filled for a given volume of the cooler?

A: No, this equation is only applicable for finding the number of cups that can be filled for a given capacity of each cup. If you want to find the number of cups that can be filled for a given volume of the cooler, you will need to use a different equation.

Q: What are some real-world applications of this equation?

A: This equation has many real-world applications, such as:

Calculating the number of cups that can be filled for a given capacity of each cup in a restaurant or bar.
Determining the number of cups that can be filled for a given capacity of each cup in a sports event or festival.
Calculating the number of cups that can be filled for a given capacity of each cup in a manufacturing process.

Q: Can I use this equation to find the capacity of each cup for a given number of cups?

A: Yes, you can use this equation to find the capacity of each cup for a given number of cups. To do this, you can rearrange the equation to solve for $x$ :

x = \frac{k}{c}

where $x$ is the capacity of each cup, $k$ is the constant of variation, and $c$ is the number of cups.

Q: What are some common mistakes to avoid when using this equation?

A: Some common mistakes to avoid when using this equation include:

Not using the correct value for the constant of variation.
Not using the correct value for the capacity of each cup.
Not using the correct equation for the problem at hand.