Question

The total revenue for Dante's Villas is given as the function R(x) = 500x -0.5x², where x is the number of rooms booked. What number of rooms booked produces

the maximum revenue?

Answer 1

500 rooms booked will produce the maximum revenue

Explanation:

Revenue function

`R(x) = -0.5x^2 + 500x`

where x = number of rooms booked

The maximum revenue is determined by setting the marginal revenue to 0 and solving for x

The marginal revenue is the first derivative of R(x) = R'(x)

`MR = R'(x) =`
`(d)/(dx)\left(-0.5x^2+\:500x\right)`

`=-(d)/(dx)\left(0.5x^2\right)+(d)/(dx)\left(500x\right)`

`(d)/(dx)\left(0.5x^2\right)= = 2(0.5x) = x\\\\(d)/(dx)\left(500x\right)=500`

`MR = -x + 500`

Set this equal to zero and solve for x

`-x + 500 = 0\\\\x = 500\\\\`

So revenue is maximized at 500 rooms booked

The maximum revenue is found by substituting 500 in the original equation for revenue

`R\left(500\right)=-0.5\left(500\right)^2+\:500\left(500\right)\\\\= 375,000`