A rubber band is strung around six circles (each with radius r cm).
a) Show that the area of the shape is given by the expression r^2(12+6√3+π).
b) Show that the total length of the rubber band is given by the expression 2r(6+π).
I can't picture the shape.
My bad, sorry, should have given more information.
Anyways, the shape is supposedly shaped like a hexagon (2 circles on top 2 circles on the bottom, and one circle in the left and right middle to form a hexagon). If you still can't picture it, search up "6 circles shaped like a hexagon in Google Images," and it's the 6th picture on the first row (ignore the hexagon in the middle of the picture).
Where is this phrasing "search up" coming from? Just the word "search" is all you need.
still need help :L
length of semi ellipse on one end = 2r+rsqrt3 =a
at distance r from ellipse center semi height of ellipse is r
find height of ellipse at center
r^2/[(2+sqrt3)^2r^2] + 4r^2/b^2 = 1
then
area = pi a b
at distance r from ellipse center semi height of ellipse is 2r
To solve this problem, we need to break it down into smaller parts and analyze each component separately. Let's analyze each part of the problem step by step.
a) To find the area of the shape formed by the rubber band, we need to calculate the area of each individual circle and subtract the overlapping areas.
Each circle has a radius of r, so the area of each circle is given by A = πr^2.
There are six circles forming the shape, so the total area of the six circles is 6A = 6πr^2.
However, we need to subtract the overlapping areas. When the circles overlap, they form six interior regions shaped like equilateral triangles. The side of these triangles is equal to the diameter of each circle, which is 2r.
The area of an equilateral triangle with side length L is given by A_triangle = (sqrt(3)/4) * L^2.
Since the side length of each triangle is 2r, the area of each triangle is A_triangle = (sqrt(3)/4) * (2r)^2 = sqrt(3) * r^2.
There are six overlapping triangles formed by the circles, so the total area of the overlapping triangles is 6A_triangle = 6 * (sqrt(3) * r^2) = 6√3 * r^2.
Thus, the total area of the shape formed by the rubber band is:
Area = Total area of six circles - Total area of six overlapping triangles
= 6πr^2 - 6√3r^2
= r^2(6π - 6√3)
= r^2(12+6√3+π) (using the fact that 6π = 12π and -6√3 is the same as 6√3 with a negative sign)
Therefore, the area of the shape is given by the expression r^2(12+6√3+π).
b) To find the total length of the rubber band, we need to find the circumference of each circle and add them up.
The circumference of each circle is given by C = 2πr.
There are six circles, so the total length of the rubber band is:
Total length = Sum of the circumferences of six circles
= C + C + C + C + C + C
= 6C
= 6 * 2πr
= 12πr
Therefore, the total length of the rubber band is given by the expression 2r(6+π).