Payton colored a composite shape made up semicircles whose diameters are the sides of a square. The result is a shape given below. Find the area of the shaded region (Petals) in terms of x? Calculate the area using the dimension in 2nd figure?

Question

Avery Allen · Accepted Answer

Area #=8x^2(pi-2)#

The area of half a petal is given by

#(pi r^2)/4-r^2/2 = r^2/4(pi-2)#

See the attached figure:
in light blue is represented the circle's quarter #(pi r^2)/4#
in yellow is represented the triangle's area #r^2/2#
in red is represented the difference #(pi r^2)/4-r^2/2#

(Note: the halfed petal is rotated #45^@#)

so the four petals have an area of

#8 r^2/4(pi-2)= 2r^2(pi-2)# but #r = 2x# then

Area #=8x^2(pi-2)#

Autumn Armstrong · Answer

undefined To find the area of the shaded region (petals) in terms of $ x $, we first need to understand the composition of the shape.

The shaded region consists of four semicircles whose diameters are the sides of a square. Each semicircle has a radius equal to half the side length of the square. Therefore, the area of one semicircle is $ \frac{\pi}{2} \times \left(\frac{x}{2}\right)^2 $.

Since there are four identical semicircles, the total area of the shaded region is $ 4 \times \frac{\pi}{2} \times \left(\frac{x}{2}\right)^2 $.

Now, let's calculate the area using the dimensions provided in the second figure.

In the second figure, the side length of the square is $ x $. Therefore, the radius of each semicircle is $ \frac{x}{2} $.

Using the formula for the area of a semicircle ($ \frac{\pi}{2} r^2 $), where $ r $ is the radius:

$$ \text{Area of one semicircle} = \frac{\pi}{2} \times \left(\frac{x}{2}\right)^2 $$

$$ = \frac{\pi}{2} \times \frac{x^2}{4} $$

$$ = \frac{\pi x^2}{8} $$

Since there are four identical semicircles, the total area of the shaded region is:

$$ \text{Total area} = 4 \times \frac{\pi x^2}{8} $$

$$ = \frac{\pi x^2}{2} $$

So, the area of the shaded region (petals) in terms of $ x $ is $ \frac{\pi x^2}{2} $.