Question

EC 5 pts. A campground rents campsites for $12 per night. At this rate, all

90 campsites are usually rented. For each $1 increase in the price
per night, about 3 less sites are rented. The campground's nightly
revenue can be modeled by R=(90 - 3x)(12 + x). Use the
vertex form to find how the campground can maximize nightly
revenue

Answer 1

To find the maximum nightly revenue for the campground, we need to use the vertex form of the quadratic equation. The vertex form is given by the equation R = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.

In this case, we have R = (90 - 3x)(12 + x).

To rewrite this equation in vertex form, we need to expand and simplify the expression:

R = (90 - 3x)(12 + x)

= 1080 + 90x - 36x - 3x^2

= -3x^2 + 54x + 1080

Now, let's rewrite the equation in vertex form by completing the square:

R = -3(x^2 - 18x) + 1080

= -3(x^2 - 18x + 81) + 1080 + 3(81)

= -3(x - 9)^2 + 1293

The equation is now in vertex form, R = -3(x - 9)^2 + 1293. Comparing this equation to the vertex form equation R = a(x - h)^2 + k, we can see that the vertex of the parabola is located at the point (9, 1293).

Therefore, to maximize the nightly revenue, the campground should set the price per night at $9, which will result in a maximum revenue of $129

alli <3