Make q the subject of the relation t = $\sqrt{(\frac{pq}{r} - r^2q)}$

Analysis

The goal is to isolate $q$ in the given equation: $t = \sqrt{\frac{pq}{r} - r^2q}$

Concept

To make a variable the subject of a formula, we need to perform algebraic operations (squaring, factoring, and dividing) to isolate that variable on one side of the equation.

Step-by-Step Solution

Step 1: Eliminate the square root Square both sides of the equation to remove the radical: $t^2 = \frac{pq}{r} - r^2q$

Step 2: Clear the fraction To make the equation easier to handle, multiply every term by $r$ to eliminate the denominator: $rt^2 = pq - r^3q$

Step 3: Factor out $q$ Since $q$ is common to both terms on the right side, factor it out: $rt^2 = q(p - r^3)$

Step 4: Isolate $q$ Divide both sides by $(p - r^3)$ to get $q$ by itself: $q = \frac{rt^2}{p - r^3}$

Conclusion

Comparing our result to the given options, we find that it matches option A.

Exam Tip

When dealing with formulas involving square roots, always square both sides first to simplify the expression. Also, whenever you see a variable appearing in multiple terms on one side, factoring is almost always your next logical step to isolate that variable!