[f4 maths]Functions

a)
Since for every $1 increase in the ticket price, the number of tickets sold will decrease by 50, the relation between p and q should be a linear equation.

Let q = mp + c
Put p=160, q=12 000, we have:
12 000 = 160m + c ----------------(1)
Put p = 161, q = 11 950, we have:
11 950 = 161m + c ----------------(2)

(2) - (1):
-50 = m
Put m= -50 into (1):
12 000 = 160*(-50) + c
c = 20 000

Therefore, q = -50p + 20 000.

b)
Income = pq = p(-50p + 20 000) = -50p² + 20 000p
To get the maximum income, we use the method of completing the square:
-50p² + 20 000p
= -50(p² - 400p)
= -50(p² - 400p + 200²) + 50*200²
= -50(p - 200)² + 2 000 000
Therefore, income will be maximum when p = 200. At that time, the income is 2 000 000. In other words, the price of the tickets should be $200 in order to have maximum income.

2006-11-13 20:54:45 補充：
補充：由於票價每加$1，購票觀眾就會少50人，因此可以知道票價(p)與購票觀眾(q)是一個線性關係，因此我設q = mp + c(所有線性關係都可以寫成y = mx + c)。

上面2位仁兄, 可唔可以解釋點解係q = -50p + 20 000 o既, 我睇唔明

您好！

a) q = 20000 - 50p

b)the maximum income should be pq.
Therrefore,
pq = p(20000 - 50p)
=-50p^2 + 20000p
Sicne it is a quadratic function ,we can find the maximum volume by completing square
=-50(p^2 - 400p)
=-50(p - 200)^2+ 50(200)^2
=-50(p - 200)^2+ 2000000
that means 2000000 is the maximum volue of pq when p = 200

Therefore, $200 should the price of each ticket be in order to get the maximum income.

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