A piece of string fits exactly once around the perimeter of a square whose area is 144.  Rounded to the nearest whole number, what is the area of the largest circle that can be formed from the piece of string?
Since the area of the square is 144, each side has length $\sqrt{144}=12$.  The length of the string equals the perimeter of the square which is $4 \times 12=48$.  The largest circle that can be formed from this string has a circumference of 48 or $2\pi r=48$.  Solving for the radius $r$, we get $r=\frac{48}{2\pi} = \frac{24}{\pi}$. Therefore, the maximum area of a circle that can be formed using the string is $\pi \cdot \left( \frac{24}{\pi} \right)^2 = \frac{576}{\pi} \approx \boxed{183}$.
183
Level 5
Prealgebra
Problem: A piece of string fits exactly once around the perimeter of a square whose area is 144.  Rounded to the nearest whole number, what is the area of the largest circle that can be formed from the piece of string?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the area of the largest circle that can be formed from the piece of string, we first need to determine the length of the string. The area of the square is given as 144. The formula for the area of a square is A = s^2, where s is the length of a side. 

Calculating the side length:
s = √144 = 12.

Next, we calculate the perimeter of the square, which is given by the formula P = 4s. Thus, the perimeter is:
P = 4 * 12 = 48.

This means the length of the string is 48 units. 

Now, we can use this string to form a circle. The circumference of a circle is given by the formula C = 2πr, where r is the radius. Setting the circumference equal to the length of the string, we have:
2πr = 48.

To find the radius, we solve for r:
r = 48 / (2π) = 24 / π.

Next, we calculate the area of the circle using the formula A = πr^2:
A = π(24/π)^2 = π(576/π^2) = 576/π.

To find the numerical value of the area, we approximate π as 3.14:
A ≈ 576 / 3.14 ≈ 183.44.

Rounding this to the nearest whole number gives us 183. Thus, the area of the largest circle that can be formed from the piece of string is 183.<|END_OF_LATENT|> The final answer is $183$.