Quadratic functions應用題（10分）

[匿名]

2012-09-15 5:04 am

(f4-f5程度）
Andy is going to hold a concert at the Hong Kong Coliseum.Suppose the auditorium can accommodate at most 12000 people. He estimates that if the price for each ticket is $160, then all the tickets will be sold. For every increase of $1 in the ticket price, the number of tickets sold will decrease by 50.

(a) Let $p be the price of each ticket and q be the number of tickets sold. Express q in terms of p.
(b)What should the price of each ticket be in order to get the maximum income?

thx!!><

回答 (1)

Godfrey

2012-09-16 5:05 pm

✔ 最佳答案

(a) Let I the increase in price
P = price of each ticket
I = p - 160 --------------- (i)
Let the number of tickets sold = q
q = 12000 -50I
50I = 12000 - q
I = (12000 – q)/50 -------- (ii)

Increase in price from (i) and (ii)

P – 160 = (12000 – q)/50
50p – 8000 = 12000 – q
q = 12000 + 8000 – 50

q = 20000 – 50p (Answer)

(b) Let R be the income
Income = price of each ticket x number of tickets sold
R = p x q
R = p(20000 – 50p)
R = 20000 p – 50p^2
R = -50p^2 + 20000p

This is quadratic equation (ax^2 + bx + c = 0)
a = -50, b = 20000, c = 0
Since a is a negative number, the quadratic equation shows a maximum at the vertex
The vertex occurs when p = -b/2a (maximum value of R)
Maximum income when p = -20000/[(2)(-50)] = 20000/100 = 200

The price of each ticket should be $200 to get the maximum income (Answer)

2012-09-16 09:11:03 補充：
Correction:

q = 12000 + 8000 – 50 (wrong) ~ missing the p after 50

It should be
q = 12000 + 8000 – 50p
q = 20000 – 50p (Answer)

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