Circle C has radius x and circumference y. Quantity A: The area of a square region with side x. Quantity B: The area of a square region with perimeter y. A) Quantity A is greater.... Circle C has radius x and circumference y. Quantity A: The area of a square region with side x. Quantity B: The area of a square region with perimeter y. A) Quantity A is greater. B) Quantity B is greater. C) The two quantities are equal. D) The relationship cannot be determined from the information given. Read more

Steps to Solve

1. Identify the radius and circumference of circle C

Let the radius of circle C be represented as $r$. The circumference of circle C can be calculated using the formula for circumference, which is:

$$ C = 2\pi r $$

2. Express the area of the square based on the radius

The area of a square is given by the side length squared. If we use the radius $r$ as the side length, the area $A_r$ of the square based on the radius is:

$$ A_r = r^2 $$

3. Express the area of the square based on the circumference

For the square based on the circumference, we first need to express the side length. Since the square's perimeter is equal to the circumference of circle C, the side length, $s$, can be derived as follows:

$$ s = \frac{C}{4} = \frac{2\pi r}{4} = \frac{\pi r}{2} $$

Now, the area $A_c$ of the square based on the circumference can be calculated as:

$$ A_c = s^2 = \left(\frac{\pi r}{2}\right)^2 = \frac{\pi^2 r^2}{4} $$

4. Compare the two areas

Now that we have both areas, we can compare them as follows:

$$ A_r = r^2 $$

$$ A_c = \frac{\pi^2 r^2}{4} $$

To compare, consider the ratio:

$$ \frac{A_r}{A_c} = \frac{r^2}{\frac{\pi^2 r^2}{4}} = \frac{4}{\pi^2} $$

5. Evaluate the comparison

To know which area is larger, we determine if $ \frac{4}{\pi^2} $ is greater than or less than 1. Since $\pi \approx 3.14$, we have:

$$ \pi^2 \approx 9.86 $$

This gives us:

$$ \frac{4}{9.86} < 1 $$

Thus, $A_r < A_c$, meaning the area of the square based on the circumference is larger than the area of the square based on the radius.