This is for SAT prep :(
Semicircular arcs AB,AC,BD,CD divide the circle above into regions.
The points show along the diameter AD divide it into 6 equal parts. If AD=6, what is the total area of the shaded regions?
a.4pi
b.5pi
c.6pi
d.12pi
e.24pi
Here is a picture:
i54 . tinypic . com/25tiwib. jpg
remove the spaces
Thanks!
6pi
To find the total area of the shaded regions, we need to determine the area of each shaded region and then sum them up.
First, let's start by finding the area of one shaded region. Since the circle is divided into 6 equal parts along the diameter AD, each shaded region will be an isosceles triangle with a base AD and height equal to the radius of the circle.
We know that AD = 6, so the radius of the circle is half of that, which is 3. The area of an isosceles triangle is given by the formula A = (1/2) * base * height. In this case, the base is AD = 6 and the height is the radius = 3. Plugging these values into the formula, we get A = (1/2) * 6 * 3 = 9.
Therefore, the area of one shaded region is 9 square units.
Now, let's count the number of shaded regions in the picture. From the picture, we can see that there are 4 shaded regions.
To find the total area of the shaded regions, we multiply the area of one shaded region by the number of shaded regions: 9 * 4 = 36.
So, the total area of the shaded regions is 36 square units.
Since the area of a circle is given by the formula A = pi * radius^2, and we know that the radius is 3, we can calculate the total area of the circle. A = pi * 3^2 = 9pi.
Now, subtract the total area of the shaded regions from the total area of the circle to find the area of the unshaded region: 9pi - 36.
Comparing the answer choices, we can see that the closest option is d. 12pi.
Therefore, the answer is d. 12pi.