Question

The total revenue for Jane's Vacation Rentals is given as the function R(x) = 400x - 0.5x^2, where x is the number of villas filled. What number of villas filled produces the maximum revenue?

A) R(x) = 200x
B) R(x) = 400x^2
C) R(x) = 400x - 0.5
D) R(x) = 0.5x^2 - 400x
E) R(x) = 0

Answer 1

The maximum total revenue for Jane's Vacation Rentals occurs when 400 villas are filled, as calculated by finding the x-coordinate of the vertex of the given quadratic revenue function.

Step-by-step explanation:

To find the number of villas filled that produces the maximum total revenue for Jane's Vacation Rentals, we use the given revenue function, R(x) = 400x - 0.5x2. The maximum value of this quadratic function occurs at its vertex. Since the quadratic is in the form of R(x) = ax2 + bx + c, we can find the x-coordinate of the vertex using the formula -b/(2a). Here, a = -0.5 and b = 400.

Calculating the x-coordinate of the vertex, we get:

• x = -b / (2a)
• x = -400 / (2(-0.5))
• x = -400 / (-1)
• x = 400

Therefore, filling 400 villas will produce the maximum revenue for Jane's Vacation Rentals. Note that since 'x' represents the number of villas, it should be a whole number. As the quadratic equation gives us a maximum at x = 400, and because we cannot fill a fraction of a villa, the actual number of filled villas would be either 399 or 400, depending on additional context not provided in the question.