The problem requires rearranging the given formula for $d$ in terms of the other variables. The formula provided is:

$A = \frac{\pi}{4} \cdot \left( D^2 - d^2 \right)$

Here’s the step-by-step solution to isolate $d$:

Step 1: Eliminate the fraction

Multiply both sides by $\frac{4}{\pi}$ to remove $\frac{\pi}{4}$:

$\frac{4A}{\pi} = D^2 - d^2$

Step 2: Rearrange for $d^2$

Move $D^2$ to the other side:

$d^2 = D^2 - \frac{4A}{\pi}$

Step 3: Solve for $d$

Take the square root of both sides:

$d = \sqrt{D^2 - \frac{4A}{\pi}}$

Final Answer:

$d = \sqrt{D^2 - \frac{4A}{\pi}}$

This matches the third solution displayed on the image. The other steps show algebraic transformations leading to the same result.

Would you like further clarification or explanations about any step? Let me know!

Related Questions:

1. How can the same equation be rearranged to solve for $D$ instead?
2. What happens to $d$ when $A = 0$?
3. Can the square root in the solution result in imaginary numbers? When?
4. How would you solve the equation if $A$ or $\pi$ were negative?
5. Can this formula be graphed for $d$ as a function of $A$ and $D$?

Tip:

Always simplify equations step-by-step by isolating terms and eliminating fractions before solving for the unknown variable.