急!F6 Geometric Sequences 02ex2

Question 2

(a)

Let $a be the profit in January.

It is given that
a - 3x = 32000 ...[1]
a - 6x = 14000 ...[2]

[1]-[2] gives 3x = 18000
x = 6000
a = 32000+18000 = 50000

The value of x is 6000.

The profit in January is $50000.

(b)
The arithmetic sequence starts with the first term 50000 with common difference -6000.

Consider T(6) = 50000 + 5*(-6000) = 20000

S(6) = [T(1) + T(6)]*6/2 = 210000

The total profit is $210000.

(c)
Consider T(n) < 0

50000 + (n-1)*(-6000) < 0

50 + (n-1)*(-6) < 0

50 - 6n + 6 < 0

6n > 56

n > 9.33333333...

That is, starting from October, the shop starts to make a loss.


Question 3

(a)
Note that [(2n-1)+1]/2 = n is the number of rods from the first longest to the shotest.

Consider an arithmetic sequence with first term being 160 and common difference -4.

T(n) = 32

160 + (n-1)(-4) = 32

160 - 4n + 4 = 32

4n = 132

n = 33

(b)
From the first rod of 160 cm to the 33th rod of 32 cm, the total length is
(160+32)*33/2 = 3168 cm

For the bottom part, there are 2*33-1-33 = 32 rods, starting from 36 cm to 160 cm, the total length is
(36+160)*32/2 = 3136 cm

Therefore, the total length is 6304 cm.