Six pipes each having a radius of 0.5 feet are stacked in a triangular pile with three pipes on the ground tangent to each other, two in the next row and one on top. What is the height of the pile. Express you answer in simplest radical form
A good idea is to take 6 identical coins and line them up as above (6 quarters)
draw a line from the centre of the bottom middle circle to the centre of the left circle of the middle row.
Draw a horizontal line from the centre of the left circle in the second row to the point of tangency with the other circle in that row.
Draw a vertical from that point of tangency to the centre of the bottom middle circle.
You should have a right-angled triangle with hypotenuse 1, short leg as 0.5 and vertical leg as x
x^2 + .5^2 = 1^2
x^2 = .73 or 3/4
x = √3/2
so the distance from the outside of the bottom middle circle to the centre of the 6 circles is
√3/2 + .5
Then the height of the whole stack is
2(√3/2 + .5)
= √3 + 1
In a particular game, a fair die is tossed. If the number of spots showing is either four or five, you win $1. If the number of spots showing is six, you win $4. And if the number of spots showing is one, two, or three, you win nothing. You are going to play game twice.
The probability that you win $4 both times is
a)1/6
b)1/3
c)1/36
d)1/4
e)1/12
1/3
To find the height of the pile of pipes, we can use the concept of an equilateral triangle.
First, let's find the height of one pipe. Since the radius of each pipe is 0.5 feet, the diameter is 2 times the radius, which is 1 foot. Therefore, the height of one pipe is equal to the radius, which is 0.5 feet.
Now, let's consider the triangular pile of pipes. The triangular arrangement forms an equilateral triangle, meaning all the sides have equal length.
The height of an equilateral triangle can be found using the formula:
height = (√3/2) * side length
In our case, the "side length" of the equilateral triangle is equal to the sum of the heights of the three pipes at the bottom row, which is 0.5 + 0.5 + 0.5 = 1.5 feet.
So, the height of the pile (equilateral triangle) can be calculated as:
height = (√3/2) * 1.5
= (√3/2) * 3/2
= (3√3)/4 feet.
Therefore, the height of the pile is (3√3)/4 feet, expressed in the simplest radical form.