three circles are externally tangent to each other. The radius of each circle is 6 inches. Find the following:

a)the length of the shortest wire athat goes around them.

b) the shaded area (space between them when stacked one circle on top of two)

a) If we draw lines connecting the centers of the circles, we will form an equilateral triangle with side length 12 inches (since the radius of each circle is 6 inches). To find the length of the shortest wire, we need to find the perimeter of a "hexagon" formed by the three parts of three circles.

First, find the length of a segment of each of the three circles that are not covered by the wire. Since the triangle formed by the centers of the circles is equilateral, the angle between any two of these line segments is 60 degrees. Therefore, each exposed part of circle is one-sixth of the circumference. The circumference of each circle is 2*pi*6 = 12*pi inches, so one-sixth of that is 2*pi inches. Therefore, the length of the wire covering the remaining part of each circle is:

12*pi - 2*pi = 10*pi inches

Since there are three exposed parts of circles, the total length of the wire is:

3 * (10*pi) = 30*pi inches

b) To find the shaded area, we can find the area of the equilateral triangle formed by the centers of the three circles and subtract the area of the three circle sectors that are inside the triangle.

To find the area of the equilateral triangle, we can use its side length and the formula:

Area = (s^2 * sqrt(3)) / 4

Where s is the side length. In this case, s = 12 inches, so:

Area = (12^2 * sqrt(3)) / 4 = (144 * sqrt(3)) / 4 = 36*sqrt(3) square inches

Next, find the area of the three circle sectors. Each sector has a central angle of 60 degrees, since that is the angle between the sides in the equilateral triangle:

Sector area = (angle / 360) * pi * r^2

Where angle is the central angle and r is the radius of the circle. In this case, angle = 60 and r = 6 inches, so:

Sector area = (60 / 360) * pi * 6^2 = (1/6) * pi * 36 = 6*pi square inches

Since there are three sectors, the total area of the sectors is:

3 * (6*pi) = 18*pi square inches

Finally, subtract the area of the sectors from the area of the triangle to find the shaded area:

Shaded area = 36*sqrt(3) - 18*pi square inches

To find the length of the shortest wire that goes around the three externally tangent circles, we can utilize the fact that the line connecting the centers of two externally tangent circles is perpendicular to the tangent line that joins them.

a) The three circles are externally tangent to each other, forming a triangle with sides equal to the diameters of the circles. Since the radius of each circle is 6 inches, the diameter is 2 * 6 = 12 inches. Now, we have an isosceles triangle with two sides of length 12 inches and a base of 12 inches.

To calculate the length of the shortest wire that goes around the circles, we can find the perimeter of this triangle. The perimeter of an isosceles triangle can be calculated by multiplying the length of the base by 2 and adding it to the length of the other two sides. In this case, the perimeter would be 12 + 12 + 12 = 36 inches.

b) To find the shaded area between the circles when they are stacked one circle on top of two, we need to calculate the area of one circle and subtract it from the area of the three circles combined.

The area of one circle can be calculated using the formula: A = π * r^2, where r is the radius of the circle. In this case, r = 6 inches. Thus, the area of one circle is A = π * 6^2 = 36π square inches.

The total area of the three circles combined is 3 * 36π = 108π square inches.

To find the shaded area, we subtract the area of one circle from the total area of the three circles: Shaded area = 108π - 36π = 72π square inches.