The total revenue for Fred's Estates LLC is given as the function R(x)=200x−0.25x2
R
(
x
)
=
200
x
−
0.25
x
2
, where x is the number of villas booked. What number of villas booked produces the maximum revenue?
R(x)= 200x - 0.25x^2
quick way to find vertex:
the x of the vertex is -b/(2a) = -200/(-.5) = 400
R(400) = 200(400) - .25(400)^2 = 40,000
To find the number of villas booked that produces the maximum revenue, we need to determine the maximum point of the revenue function R(x) = 200x - 0.25x^2.
To do this, we can use calculus and find the critical points of the function. The critical points occur where the derivative of the function is equal to zero or is undefined.
First, let's find the derivative of the revenue function R(x) with respect to x:
R'(x) = d/dx [200x - 0.25x^2]
= 200 - 0.5x
Now, to find the critical points, we set the derivative equal to zero and solve for x:
200 - 0.5x = 0
Solving for x, we get:
0.5x = 200
x = 400
The critical point is x = 400.
To determine whether this critical point is a maximum or minimum, we can use the second derivative test.
Let's find the second derivative of R(x):
R''(x) = d^2/dx^2 [200 - 0.5x]
= -0.5
Since the second derivative is negative, this means that the function R(x) is concave down. Therefore, the critical point x = 400 corresponds to the maximum revenue.
Thus, when 400 villas are booked, it will produce the maximum revenue for Fred's Estates LLC.
To find the number of villas booked that produces the maximum revenue, we need to determine the value of x that maximizes the function R(x).
Step 1: Take the derivative of the revenue function R(x) with respect to x.
R'(x) = 200 - 0.5x
Step 2: Set the derivative equal to zero and solve for x.
200 - 0.5x = 0
Simplifying the equation:
0.5x = 200
Dividing both sides by 0.5:
x = 400
Therefore, the number of villas booked that produces the maximum revenue is 400.