F.4 Maths Function questions

既然是 F.4 數，應該不知郭老師的 part 1b 在做甚麼的。
我嘗試用你剛學完的 quadratic function 計算吧。
1a. Area of lawn is. :
(π/4) x^2 + 2x(10 - x) + (π/4)(10 - x)^2
= (π/2 - 2)x^2 + (20 - 5π)x + 25π
= (1/2)(π - 4)x^2 - 5(π - 4)x + 25π
= (1/2)(π - 4)(x^2 - 10x) + 25π
= (1/2)(π - 4)(x^2 - 10x + 25) + 25π - (25/2)(π - 4)
= (1/2)(π - 4)(x - 5)^2 + (25/2)(π + 4)

b. The max. of the lawn occurs when x = 5,
ie, max. area of the lawn is (25/2)(π + 4) m^2.

2. arc length (s) = rθ, area of sector = (1/2)r^2 θ = (1/2)rs
As OA : OB = 3 : 5, so OA : AB = 3 : 2, AB = 2(1 - x)
Therefore, OA = 3(1 - x) and OB = 5(1 - x)

a, A(x) = (1/2)(OB)(5x) - (1/2)(OA)(3x)
ie. A(x) = 8x(1 - x)

b. As there is no zero area, so the domain of A(x) : 0 < x < 1

c. A(x)
= 8x(1 - x)
= -2(4x^2 - 4x + 1) + 2
= -2(2x - 1)^2 + 2
Therefore, the max. area of ABCD occurs when x = 0.5,
ie. the max. area is 2 cm^2.

OA : 2(1-x) = 3 : 2
OA = 3(1-x)

Then, you can solve the angle.

Sorry! I don't know how to do Q2.


圖片參考：http://imgcld.yimg.com/8/n/HA00388954/o/20131027011506.jpg


2013-10-27 02:01:45 補充：
多謝提點，咁就易好多啦！等同學仔自己試試再算啦！

2013-10-27 02:54:10 補充：
第 2 題

a : A = (16x)(1-x) = 16x - 16x²

b : 0<= x <= 1

c : 4 square unit

如果仲係計唔到，在補充再問。

2013-10-28 12:46:29 補充：
那些年 ，多謝提醒，對提問者的幫助很大，

送佛送到西，YTC 留意!

下面 :

面積== (1/2)(π - 4)(x - 5)^2 + (25/2)(π + 4) 中

(π - 4) 為負數，因此，最大面積時是 (1/2)(π - 4)(x - 5)^2 = 0，即當x=5。

此外 arc length (s) = rθ 中 θ 是 radian 而不是 degree。