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
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)
1 answer
1 answer