The diagram is formed from 6 identical closed-packed circles. If the height of the figure is 2 feet, find the radius of each circle in inches. Show your complete solution with formula.
circles can be packed in many ways. If they are in a rectangular lattice then they may be in a 2x3 array, in which case, each circle's diameter is 1 foot.
Surely by now you have made some attempt to solve the problem...
4.39in
To find the radius of each circle in inches, we need to convert the height from feet to inches and then use the formula for the radius of a circle in a closed-packed diagram.
Given:
- Height of the figure = 2 feet
First, let's convert the height from feet to inches:
1 foot = 12 inches
So, the height of the figure in inches = 2 feet * 12 inches/foot = 24 inches
Now, let's proceed with the formula for the radius of a circle in a closed-packed diagram.
In a closed-packed arrangement, each circle touches its neighboring circles at six points, forming a hexagonal pattern. The vertical distance between the centers of two neighboring circles is equal to twice the radius of the circle.
Let's assume that the radius of each circle is "r" inches. In the closed-packed diagram, the vertical distance between the centers of the circles is equal to 2 times the radius, which means it is 2r.
We are given the height of the figure, which is equal to the sum of the vertical distances between the centers of the circles. In our case, the height is 24 inches.
Therefore, we can set up the equation:
2r + 2r + 2r + 2r + 2r + 2r = 24
Simplifying the equation, we get:
12r = 24
Dividing both sides by 12, we solve for the radius:
r = 24/12
r = 2 inches
Hence, the radius of each circle in inches is 2 inches.