An important factor in solid missile fuel is the particle size distribution. significant problems occur if the particle sizes are too large. from production data in the past, it has been determined that the particle size (in micrometers) distribution is characterized by 3x−4, x > 1, f(x) = 0, elsewhere. (a) verify that this is a valid density function. (b) evaluate f(x). (c) what is the probability that a random particle from the manufactured fuel exceeds 4 micrometers?

b)

c) 0.0156

Step-by-step explanation:

Let's call X : '' Particle size (in micrometers) ''

X is a random variable

The distribution function for X is :

For x > 1

f(x) = 0, elsewhere

For a) the condition for f(x) to be a valid density function is that

the integral between -∞ and ∞ of f(x) must be equal to 1.

For the integral I change ∞ for j so ∞ = j

Also for the integral I change -∞ for k so k = -∞

Then f(x) is a valid density function

b) To find F(x) we must integrate between -∞ and x the function f(t)

We calculate f(t) changing x by t in the f(x) function

Now we integrate

c) P(X>4)

P(X>4) = 1 - P(X≤4)

P(X>4) = 1 - F(4)

80π/3 or about 83.775

step-by-step explanation:

since a cone is 1/3 of a cylinder, we need to find the base's area. since area = πr^2, we get 16π. then, multiplying this by 5, we get 80π. dividing by 3, since the cone's volume is 1/3 of that of a cylinder's, we get 80π/3.

hope this !

your answer would be 12