The equation A = pi r^2 relates the area A of a circle to its radius r. Solve the formula for r, then find the radius of a circle with area 78.5 square centimetres, using 3.14 for pi.

Step-by-step solution

1. 1

   Undo the multiplication before the power. $r$ is squared and multiplied by $\pi$. Peel the operations off in reverse order: divide by $\pi$ first, take the square root second. Taking the root first would strand a $\sqrt{\pi}$ in the wrong place.

2. 2

   Divide both sides by $\pi$.

   $\frac{A}{\pi} = r^2$

3. 3

   Take the square root and keep only the positive root. Algebraically $r = \pm\sqrt{A/\pi}$, but a radius is a length, so the negative root is discarded on physical grounds:

   $r = \sqrt{\frac{A}{\pi}}$

4. 4

   Substitute the numbers. With $A = 78.5$ square centimetres and $\pi \approx 3.14$:

   $r = \sqrt{\frac{78.5}{3.14}} = \sqrt{25} = 5$

   The division comes out to exactly $25$, which is a strong hint the problem was designed around $\pi = 3.14$ — that is why the question specifies it.

5. 5

   Check forwards. Area from $r = 5$: $3.14 \times 5^2 = 3.14 \times 25 = 78.5$ square centimetres ✓, matching the given area exactly. Using the true $\pi = 3.14159\ldots$ instead would give $r = \sqrt{78.5/\pi} \approx 4.9987$ cm, so the neat $5$ depends on the stated approximation.

6. 6

   State both answers. Formula: $r = \sqrt{A/\pi}$. Radius: $5$ centimetres.