急~~高一數3題

Q1: 本金 1000 元, 月利率 0.8%, 每月計算複一次, 求一年後的本利和? (精確到1)

Sol:

1000 * ( 1 ＋ 0.8% )^12

＝ 1000 * 1.008^12

≒ 1000 * 1.100

≒ 1100

Ans: 1100 元

Q2: 年利率 10%, 每半年計算複利一次, 希望在 4 年後得到本利和 10000元,

求本金應為多少?

Sol:

Let x be the original deposite

x * ( 1 ＋ 10% / 2 )^8 ＝ 10000

x * 1.05^8 ＝ 10000

x ＝ 10000 / 1.05^8

x ≒ 6768

Ans: 6768 元

Q3: 建築一個容積為 8000 m^3, 深為 6 m 的長方體蓄水池, 池壁造價為

a 元 / m^2, 池底的造價為 2a 元 / m^2, 把總造價 y ( 元 ) 表示為底的一邊長

x ( m ) 的函數.

Sol:

Volume is 8000 m^3 and the depth is 6 m

the Area of bottom is 8000 / 6 m^2

one side of the rectangle is x m

the the another side of the rectangle is equal to 8000 / 6x m

y ＝ 2a * 8000 / 6 ＋ a * ( 2 * 6 * x ＋ 2 * 6 * 8000 / 6x )

y ＝ 16000a / 6 ＋ a * ( 12x ＋ 16000 / x )

y ＝ 16000a / 6 ＋ 12ax ＋ 16000a / x

y ＝ 12ax ＋ 8000a / 3 ＋ 16000a / x

Ans: y ＝ 12ax ＋ 8000a / 3 ＋ 16000a / x

Let X be one side of the rectangle
and Z be the other side of the rectangle.
8000 = 6 X Z
=> Z = 8000/6X ------------ (1)

Y = (2)(a)(6X + 6Z) + (2a)(XZ) -------------- (2) (cost of the construction)
substitute (1) into (2) we get
Y = 2a(6X + 6(8000/6X)) + 2aX(8000/6X)
= 2a(6X + 8000/X) + 2a(8000/6)

Simply it you got the function y = f(x)

Moreover, to minimun the cost of building, you have to find out which
value of X that make (6X + 8000/X) minimun.
I guest that it should be around X = Z