https://file.aiquickdraw.com/mathbot/file/1730651096120lq0vpqi7.jpg

The expression given in the image is:

\[
t = \sqrt{\frac{p q}{r} - r^2 q} \quad \text{(make } q \text{ the subject)}
\]

To solve for \( q \) in terms of the other variables, let's work through the algebraic steps.

### Solution:

1. **Rewrite the equation**:
   \[
   t = \sqrt{\frac{p q}{r} - r^2 q}
   \]

2. **Square both sides to remove the square root**:
   \[
   t^2 = \frac{p q}{r} - r^2 q
   \]

3. **Isolate terms involving \( q \)**:
   Bring terms with \( q \) to one side:
   \[
   t^2 = q \left( \frac{p}{r} - r^2 \right)
   \]

4. **Solve for \( q \)**:
   Divide both sides by \(\frac{p}{r} - r^2\):
   \[
   q = \frac{t^2}{\frac{p}{r} - r^2}
   \]

Alternatively, you can simplify this by finding a common denominator if needed. Let me know if you would like further clarification on any of these steps.

---

Would you like additional details or further explanations on any step?

### Related Questions:

1. How would you isolate \( p \) in the same equation?
2. What happens if \( r = 0 \)? How would this affect the equation?
3. Can you derive \( q \) in terms of \( p \) and \( t \) only, eliminating \( r \)?
4. How would the solution change if the term \( r^2 q \) was instead \( r^3 q \)?
5. Can you check the dimensions or units of \( t \), \( p \), \( q \), and \( r \) to ensure consistency?

#### Tip:
When dealing with square roots, squaring both sides is a powerful tool, but be mindful of extraneous solutions introduced by this step.

Make q the subject in the equation t = sqrt((pq / r) - r^2 * q)

Learn how to rearrange the equation t = sqrt((pq / r) - r^2 * q) to make q the subject. This solution provides a step-by-step approach involving algebraic manipulation and removing square roots. Suitable for students in Grades 10-12.

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