A certain geometric figure has a​ volume, Upper A​, varies jointly with the square of w and q. The constant of proportionality is 2 pi. Write an equation for Upper A.

To write an equation for the volume \( A \) based on the given relationship, we know that the volume \( A \) varies jointly with the square of \( w \) and \( q \). This means the volume is proportional to \( w^2 \) and \( q \), with the constant of proportionality given as \( 2\pi \).

The general formula for joint variation is:
\[
A = k \cdot w^2 \cdot q
\]
where \( k \) is the constant of proportionality.

Given that \( k = 2\pi \), the equation becomes:
\[
A = 2\pi \cdot w^2 \cdot q
\]

This is the equation for the volume \( A \) in terms of \( w \) and \( q \), with a proportionality constant of \( 2\pi \).

Do you want further details or have any questions?

Here are five related questions:
1. How does changing the value of \( w \) affect the volume \( A \)?
2. How does the value of \( q \) influence the volume \( A \)?
3. What happens to \( A \) if both \( w \) and \( q \) are doubled?
4. Could you give an example of specific values for \( w \) and \( q \) and calculate \( A \)?
5. What is the significance of the constant \( 2\pi \) in this context?

**Tip:** When dealing with joint variation, always identify the variables and the proportionality constant clearly to set up the equation correctly.

Understand how the volume of a geometric figure varies jointly with the square of w and q, with a constant of proportionality of 2π. This problem involves joint variation concepts, volume calculations, and proportionality. Suitable for students in Grades 9-12.

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