Given that
The function of revenue is R(x) = 100x - 0.2x^2
Explanation -
It is given that we have to find the number of rooms filled that gives maximum revenue.
And x is the number of rooms filled and R is the total revenue.
If the [Total revenue] value of the function is maximum it means its derivative is zero so we will find the derivative first.
Derivative of given function
[tex]\begin{gathered} \frac{dR(x)}{dx}=\frac{d}{dx}(100x-0.2x^2) \\ R^{\prime}(x)=100-0.2\times\times2x=100-0.4x \end{gathered}[/tex]Since derivative of R(x) is zero then,
100 - 0.4x = 0
0.4x = 100
x = 100/0.4 = 1000/4 = 250
So the number of rooms filled to give maximum revenue is 250.
The final answer is 250.