急!F6 Geometric Sequences 02q13

29.

a, ar, ar², ar³ ........
250000, 225625, 214344

a = 250000
r = 0.95
Sn = a(1-r^n)/(1-r)
S(∞) = a/(1-r)

a)

1st month 250000
2nd month 250000X0.95 = 237500
3rd month 237500X0.95 = 225625
4th month225625X0.95 = 214344
5th month214344X0.95 = 203627
6th month203627X0.95 = 193445

First half year = 1074541 barrels

b) Sn = a(1-r^n)/(1-r)
2000000 = 250000(1-0.95^n)/(1-0.95)
8 = (1-0.95^n)/0.05
0.4 = 1-0.95^n
0.95^n = 0.6
nlog(0.95) = log(0.6)
n = 9.96 = 10 months

c) S(∞) = a/(1-r)
= 250000/(1-0.95) = 5000000
The total oil production will not exceed 5000000 barrels.

30. a) XY² + XZ² = 20² + 20² = 2X20²
YX² = (20√2)² = 2X20²
∆XYZ is right angled triangle.

b) let height of ∆XYZ with base YZ be h
XY² = (YX/2)² + h²
20² = (10²)(2) + h²
h = √200 cm

A1 = (1/2)[(1/2)(20√2)](√200)
A1 = 100 cm²

A2 = (1/4)(100) = 25 cm²
A3 = (1/4)(25) = 6.25 cm²

c) An = (100)(1/4)^(n-1)
A(n+1) = (100)(1/4)^n

d) S(∞) = a/(1-r)
A1+A2+A3+.... = (100)/[1-(1/4)]
= 133.33 cm²


2013-10-23 18:32:31 補充：
Thank You very much. 由於已經有其他人作答，提問者可以參考他的答案，不須修改了。

2013-10-24 00:36:59 補充：
Sorry! A1中的高計錯了，大了２倍，A1應該為 50 cm²，而A2＝12.5 cm²，A3＝6.25 cm²

c) A(n+1) = (50)(1/4)^n

d) = (50)/[1-(1/4)]
= 66.67 cm²

2013-10-24 00:38:48 補充：
A3 = 3.125 cm²

郭老師，29a 你漏了第一個月。另外，題目要求
correct to ten thousand
"ten thousand" 中文是 "萬", 002 誤會作 "十萬".

郭老師的 30 是正確的，不過我建議 30c 的答案用
A(n) = 400/4^n, A(n+1) = 400/4^(n+1) = 100/4^n

29a. 250000(1-95%^6)/(1-95%)
=250000(1-0.73509)/(0.05)
=1324550
=1300000 (corr. to the nearest ten thousand)
29b. let it will take n monthes
250000(1-95%^n)/(1-95%)=2000000
(1-95%^n)/(0.05)=2000000/250000
(1-0.95^n)=0.4
0.95^n=0.6
log(0.95^n)=log(0.6)
n log(0.95)=log(0.6)
n=log(0.6)/log(0.95)
n=9.95917
so that, it will take 10 months.
29c. let us take the infinite
250000/(1-95%)
=250000/0.05
=5000000
so , it can't exceed 5000000.

30a. XY^2+XZ^2
=20^2+20^2
=800
YZ^2
=(20(2)^0.5)^2
=800
XY^2+XZ^2=YZ^2, so ^XYZ is a right-angled triangle.

30b. A1=5*5/2=12.5
A2=2.5*2.5/2=3.125
A3=1.25*1.25/2=0.78125

30c. An = [12.5(0.5)^n]^2/2 = 200(0.5)^(2n)
An+1 = [12.5(0.5)^(n+1)]^2/2 = 200(0.5)^(2n+2)

An+1/An = (0.5)^2
= 0.25 it is a constant radio
30d. 12.5/(1-0.25)
=16.6666666666

2013-10-23 11:59:09 補充：
30b. coz ^XYZ is a right-angled triangle.
Area of ^XYZ = XY*XZ/2