Alan

Success:  p = 0.75

Fail:  q = 1 - p  = 0.25

a)  Prob of missing all 10 shots = q¹⁰ ≈ 9.5*10^-7

b) Prob of at least one success = 1 - prob of no successes = 1 - q¹⁰.

Since all cross-sections of a sphere are circles we can use the fact that  r² = 4*(16 - 4)  where r is the radius of the circular water surface.

So r = sqrt(48) = 4sqrt(3) m. Hence diameter = 8sqrt(3) m

You can also do this as follows:

EP, as a terawatt is a unit of power, then 180000 of them reaching the Earth would mean there are 180000 teraJoules of energy reaching the Earth every second.

Expand a(x+p)² + q as  ax² + 2apx + ap² + q

Compare with 3x² + 2.4x term by term:

x² term:  a = 3

x term:  2ap = 2.4

constant term: ap² + q = 0

Use these three equations to find a, p and q.

Assuming \(2^{x^2} \) means \(2^{(x^2)}\) then:

Let n = initial number of stamps

We are told:   n + 31 = 3*(n - 31)

I'll leave you to determine n from this and hence the number, (n + 31), that Cheko ended up with.

Probability = (10/50)*(9/49)*(8/48)*(8/47)*(7/46) ≈ 0.000159

This graph should help:

I assume this means that the ice cream is half in and half out of the cone, in which case:

The diameter of the cone is 6 inches (big cone!) as circumference = π*diameter.

Hence bottom of cone is 3 inches below cone opening.

Hence depth of cone = 3 + (1/3)*3 = 4 inches.

Volume of cone, Vc = (1/3)π*3² *4 in³

Volume of ice-cream hemisphere inside cone, Vi = (1/2)*(4/3)*π*3³ in³.

So volume of cone not filled with ice cream V = Vc - Vi