simple quadratic function word problem. Ten points for the first correct answer?

Question

OK, when a basketball team charges $4 per ticket, avg. attendance is 400 people. For each 20 cent decrease in ticket price, avg. attendance increases by 40 people. What should the ticket price be to maximize revenue?

THANKS SO MUCH!

Answer 1

Let x be the number of times you decrease the tickets by 20 cents.
For every x decreases in the ticket price (4 - 0.20x) you have an increase of 40x in the people (400 + 40x).

The revenue is the product of these two:
f(x) = (4 - 0.20x)(400 + 40x)

Multiply this out and you get:
f(x) = 1600 -80x + 160x - 8x²
f(x) = 1600 + 80x - 8x²
f(x) = -8(x² - 10x - 200)

If you know calculus, you can take the derivative, set it to zero and that gives you x.
(Hint: 2x - 10 = 0)

If you don't, you can turn this into vertex form by completing the square. Take the coefficient on the x term (-10), halve it (-5) and square it (25). Add and subtract this:

f(x) = -8(x² - 10x - 25 + 25 - 200)
f(x) = -8(x² - 10x - 25) + 8(225)
f(x) = -8(x - 5)² + 1800

y = a(x - h)² + k

Vertex is at (h, k) = (5, 1800)

Now you can see that the maximum amount is 1800, when (x - 5) = 0, or x = 5.

This means you will have 5 decreases of 0.20 in the ticket price, and 5 increases in the attendance of 40 people.

Ticket is $1 less ($3) and the attendance is 200 people more (600 people).

Answer: $3.

$3 x 600 people = $1800

Answer 2

$3

Answer 3

Let x = the number of 20 cent price decreases. Then the revenue R(x) is given by
R(x) = (400-40x)(4-.20x)
= -8x^2 + 80x + 1600
The maximum occurs at this parabola's axis of symmetry, that is x = -80/(2(-8)) = 5. Thus the ticket price is 4 - .20(5)= $3.

Answer 4

Let x be the amount of times the ticket price is decreased by 20 cents. And y be revenue:
y = (400 + 40x)*($4 - $0.2x) = 200 + 10x - x^2
Take the derivative of y and solve for y'=0.

y'=0=10-2x. x=5. Plug x back into ($4-$0.2x), you get $3.00

Answer 5

$4.00 x 400 = $1600
$3.80 x 440 = $1672
$3.60 x 480 = $1728
$3.40 x 520 = $1768
$3.20 x 560 = $1792
$3.00 x 600 = $1800
$2.80 x 640 = $1792 decreasing now
so $3.00 will get maximum revenue = $1800

Answer 6

$3.00