Math problems

題目已閱, 作答需時, 請耐心等候。

2014-08-13 11:56:09 補充：
(a)(1) First, we need to understand the question that what is the distance between point D
and the wall OA. It means that let a point X be on the wall OA
such that DX is perpendicular to the wall OA and we need to find DX.

∠ABO + ∠BAO + ∠AOB = 180° (angle sum of triangle)
60° + ∠BAO + 90° = 180°
∠BAO = 30°

∠BAO + ∠BAD + ∠DAX = 180° (adjacent angles on straight line)
30° + 90° + ∠DAX = 180° (interior angle of rectangle is 90°)
∠DAX = 60°

AD = BC = 45cm (opposite sides are equal)

sin∠DAX = DX / AD
sin 60° = DX / 45cm
DX = 45 sin 60° cm = (45 √3) / 2 cm = 39.0 cm (3 sig. fig.)

So, the distance between point D and the wall OA is 39.0 cm

(a)(2) Let a point Y be on the wall OB such that DY is perpendicular to the wall OB
and we need to find DY.
Also, let a point X be on the wall OA such that DX is perpendicular to the wall OA
and we need to find DX.(The point X is the same in the part (a)(1))

DY = OA + AX

sin ∠ABO = OA / AB
sin 60° = OA / 120cm
OA = 120 sin 60° cm = (120 √3) / 2 cm = 60 √3 cm

cos∠DAX = AX / AD
cos 60° = AX / 45cm
AX = 45 cos 60° cm = 45 / 2 cm = 22.5 cm

DY = (60 √3 + 22.5) cm = 126 cm (3 sig. fig.)

So, the distance between point D and the wall OB is 126 cm

(b) At this time, Θ may not be 60°.
But, we can still use related equations in part (a)(2) using Θ instead of 60°
DY = OA + AX = (120 sin Θ + 45 cos Θ) cm

In order to know which angle of Θ can make DY be the largest,
we need to integrate DY with respect to Θ.

d DY / d Θ = 0
d (120 sin Θ + 45 cos Θ) / d Θ = 0
120 cos Θ - 45 sin Θ = 0
120 cos Θ = 45 sin Θ
tan Θ = 120 / 45
Θ = 69.444° (3 d.p.)

d^2 DY / d Θ^2
= d (d DY / d Θ) / d Θ
= d (120 cos Θ - 45 sin Θ) / d Θ
= -120 sin Θ - 45 cos Θ
= - (120 sin Θ + 45 cos Θ)

d^2 DY / d Θ^2 | Θ = 69.444°
= - (120 sin 69.444° + 45 cos 69.444°)
= -128 < 0

So, Θ = 69.4° in order to make the distance between point D
and the wall OB the largest.
This largest distance = (120 sin 69.444° + 45 cos 69.444°) cm = 128 cm


2014-08-13 14:13:10 補充：
correction:
we need to integrate DY with respect to Θ
=> we need to "differentiate" DY with respect to Θ

2014-08-13 14:14:02 補充：
d is a symbol of differentiation

Did you learn differentiation?

2014-08-14 18:03:35 補充：
I think that a form 2 student is not expected to learn differentiation.
Is this question for form 2 students?

2014-08-15 07:43:56 補充：
貓貓,
我終於明白了, 謝謝!

2014-08-15 08:33:10 補充：
https://onedrive.live.com/redir?resid=A3980BFF0EA16013!217&authkey=!AAPcEcheluu7E5k&ithint=folder%2c

These two pictures have explanation and calculation. Hope you can understand.

2014-08-15 08:35:50 補充：
這次多得貓貓幫忙, 不然我也不會想得到這樣來計算,
這次最大功勞屬於貓貓的,

The greatest credit is contributed to cat cat. Thank you so much.

(b)

由於只是中二程度，所以可能不適宜用微積分 (calculus - differentiation and integration) 或三角學 (trigonometry) 的複角公式 (compound angle formula)。

不如我們先想想最遠的長度吧。
那明顯是 wardrobe 的對角線 BD，所以先用畢氏定理求出。

然後不難看到當時的 θ 就是 ∠BDA。

對於中二來說，這題是比較難。
但同時代表這是好題！

(｡◕‿◕｡)

2014-08-18 02:28:52 補充：
You all are very welcome!

╭∧---∧╮
│ .✪‿✪ │
╰/) ⋈ (\\╯

In (b), what is d

2014-08-14 13:35:11 補充：
no, please explain clearly

2014-08-14 13:55:00 補充：
this form2 question

2014-08-14 18:54:15 補充：
yes, it is

2014-08-15 12:47:38 補充：
thanks you two very much