Problem Solving

An artist wishes to paint a circular region on a square poster that is 2 feet on a side. If the area of the circular region is to be 1/2 the area of the poster, what must be the radius of the circular region in feet?

$$A.\ \frac{1}{\pi}$$
$$B.\ \sqrt{\frac{2}{\pi}}$$
$$C.\ 1$$
$$D.\ \frac{2}{\sqrt{\pi}}$$
$$E.\ \frac{\pi}{2}$$

The OA is B.

If the square has a side of 2 feet, that's mean that its area is 2 x 2 = 4.

1/2 * area the of square is = 2.

Then, we know that the area of a circle is defined as,
$$\pi\cdot r^2$$
and it should be equal 2
$$\pi\cdot r^2=2$$
$$\pi\cdot r^2=2\Rightarrow \ r^2=\frac{2}{\pi}\Rightarrow \ r=\sqrt{\frac{2}{\pi}}$$
Option B.

Experts, is there another approach to solve this PS question? Thanks!

AAPL wrote:An artist wishes to paint a circular region on a square poster that is 2 feet on a side. If the area of the circular region is to be 1/2 the area of the poster, what must be the radius of the circular region in feet?

$$A.\ \frac{1}{\pi}$$
$$B.\ \sqrt{\frac{2}{\pi}}$$
$$C.\ 1$$
$$D.\ \frac{2}{\sqrt{\pi}}$$
$$E.\ \frac{\pi}{2}$$

The OA is B.

If the square has a side of 2 feet, that's mean that its area is 2 x 2 = 4.

1/2 * area the of square is = 2.

Then, we know that the area of a circle is defined as,
$$\pi\cdot r^2$$
and it should be equal 2
$$\pi\cdot r^2=2$$
$$\pi\cdot r^2=2\Rightarrow \ r^2=\frac{2}{\pi}\Rightarrow \ r=\sqrt{\frac{2}{\pi}}$$
Option B.

Experts, is there another approach to solve this PS question? Thanks!

Hello.

Your answer is perfect.

Since the side of the square is 2, then the area is 4.

Now, the area of the circular region must be 1/2 of the square area, hence the area of the circular region must be equal to 2.

Now, we solve the equation $$\pi\cdot r^2=2\ \Rightarrow\ \ r=\pm\sqrt{\frac{2}{\pi}}.$$ Since we are talking about areas, all the numbers must be positives. Hence, the answer is $$r=\sqrt{\frac{2}{\pi}}.$$ Option B.

The side of the square is 2 feet.
Area of the poster = $$l\cdot b=2\cdot2=4feet$$
$$The\ artist\ can\ paint\ anywhere\ on\ the\ poster.\ You\ just\ have\ to\ note\ that\ area\ of\ the\ circle\ must\ be\ half\ of\ the\ square\ poster.$$
$$Area\ of\ the\ circle=\pi r^2$$
$$and\ \pi r^2\ must\ be\ =\frac{1}{2}\cdot4=2$$ $$\pi r^2=2$$ $$r^2=\frac{2}{\pi}$$
$$make\ 'r'\ the\ subject\ of\ the\ formula\ by\ squaring\ \ rooting\ both\ sides$$
$$r=\sqrt{\frac{2}{\pi}}$$