Three questions on Expected number of three heads, steinmetz solid volume, and correlation

1. What is the expected minimum number of coin tosses needed to make in order to get 3 heads in a row?
A: let x be the expected number. Consider the first toss, it has 1/2 probability to be tail, in this case, the expected number would be x+1; if the first toss is head, and the second toss is tail, this would be a probability of 1/4, with expected number x+2; if the first three are head, this would be a probability of 1/8 with expected number 3; if the first three are head, head, tail, this would be a probability of 1/8 with expected number x+3. Therefore,
x = 1/2*(x+1) + 1/4*(x+2) + 1/8*(x+3) + 1/8*3
==> x = 14.

2. Suppose 2 cylinders each with radius 1 intersect at right angles and their centers also intersect. What is the volume of the intersection?
A: a calculus solution can be obtained, see:
http://mathworld.wolfram.com/SteinmetzSolid.html
Don't know how to obtain the result with non-calculus method.

3. If X,Y, Z are 3 random variables such that X and Y are 90% correlated, Y and Z are 80% correlated, what is the minimum correlation that X and Z could have?
A: Correlation is a similar to cosine relationship. Corr(x,y) = 0.9 ==> cos(theta1)= 0.9
corr(y,z) = 0.8 ==> cos(theta2) = 0.8
min(Corr(x,z)) = cos(theta1 + theta2)
= cos(theta1)cos(theta2) - sin(theta1)sin(theta2)
= 0.72 - sqrt(1-0.9*0.9)*sqrt(1-0.8*0.8)
= 0.72 - 0.2615
= 0.4585