Find the distribution of
R=Asinθ
, where A is a fixed constant and θ is uniformly distributed on
(−π/2,π/2)
. Such a random variable R arises in the theory of ballistics. If a projectile is fired from the origin at an angle α from the earth with a speed v, then the point R at which it returns to the earth can be expressed as
R=(v2/g)sin2α
, where g is the gravitational constant, equal to 980 centimeters per second squared..

The distribution of R=Asinθ is P(θ<arcsin (t/A)) = (arcsin (t/A))π+ ½

Step-by-step explanation:

Given

To find the distribution of R=Asinθ

θ is uniformly distributed on (−π/2,π/2)

R=(v²/g)sin2α

g = 980 cm/s²

Since R=Asinθ

Divide Through by A

R/A = Asinθ/A

R/A = sinθ

So, P(R<t) = P(Asinθ<t) this is equivalent to

P(sinθ<t/A)

For t > A; P(sinθ<t/A) = 1

For |t| ≤ A; P(sinθ<t/A) = P(θ<arcsin (t/A))

For t < A; P(sinθ<t/A) = 0

So, we have

P(sinθ<t/A) = P(θ<arcsin (t/A))

P(θ<arcsin (t/A)) = (arcsin (t/A) + π/2)/π

P(θ<arcsin (t/A)) = (arcsin (t/A))/π + (π/2)/π

P(θ<arcsin (t/A)) = (arcsin (t/A))π+ ½

Hence, the distribution of R=Asinθ is P(θ<arcsin (t/A)) = (arcsin (t/A))π+ ½

The distribution of the random variable R is derived as;

[(1/π)(sin^(-1) (t/A))] + (1/2)

Step-by-step explanation:

From the question,

Random variable, R=Asinθ

So, the distribution of this variable can be represented as;

P(R < t) = P(A sin θ < t)

So let's apply conditions to it to obtain;

P(R < t) = P(A sin θ < t) {1, P(θ < sin^(-1) (t/A)), 0

At conditions of, t > A ; |t| ≤ A ; t < A; respectively.

Therefore, since uniformly distributed on (−π/2,π/2), we finally arrive at;

P(θ < sin^(-1) (t/A)) = [(sin^(-1) (t/A)) + (π/2)]/π

= [(1/π)(sin^(-1) (t/A))] + (1/2)

correct

step-by-step explanation:

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.