The mean number of riders screaming over the course of the ride is 1/4π for the density function for the number of times the riders scream on a roller coaster.
What is a function?
It is defined as a special type of relationship and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.
We have a function:
[tex]\rm f(x)=\left\{\begin{matrix}\frac{1}{4\pi}(1- \rm cos(6x) \ \ \rm if \ 0\leq x\leq 4\pi \\0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \rm otherwise\\\end{matrix}\right.[/tex]
To find the mean number of screams over the course of the ride, we will integrate the above function over the interval [0, 4π]
[tex]\rm Mean = \frac{1}{b-a} \int\limits^b_a {f(x)} \, dx[/tex]
Where a = 0 and b = 4π and [tex]\rm f(x) = {\frac{1}_{4\pi}} (1- \rm cos(6x))[/tex]
Now solving:
[tex]\rm Mean = \frac{1}{4\pi-0} \int\limits^{4\pi}_0 {{\frac{1}_{4\pi}} (1- \rm cos(6x))} \, dx[/tex]
[tex]\rm Mean = \frac{1}{4\pi} (1)[/tex] [tex](\rm \int\limits^{4\pi}_0 {{\frac{1}_{4\pi}} (1- \rm cos(6x))} \ dx =1)[/tex]
Mean = 1/4π
Thus, the mean number of screams over the course of the ride is 1/4π.
Learn more about the function here:
brainly.com/question/5245372
