SS1 Maths 3-D problems ...

2010-05-27 3:07 am

Figure: http://img192.imageshack.us/f/img192/3414/89124776.png
In the figure, OT is a lighthouse of height 250 m. O, A and B are on the same
horizontal ground. AB is a straight road of length 310 m. The angle of elevation
of the top T of the lighthouse form A and B are 48 and 62 respectively.

a) Find AT and BT. Ans: AT = 336 m, BT = 283 m

b) Find angle TAB. ans: 51.8

c) Find the angle between the plane ABT and the horizontal ground. Ans: 71.1

d) When a small object P moves along AB, is it possible that the angle of
elevation of T form P at a certain point on AB is 78 degree. Explain your
ans: no

回答 (1)

用戶2053

2010-05-27 3:34 am

✔ 最佳答案

圖片參考：http://img192.imageshack.us/img192/3414/89124776.png

a) AT = 250 / sin48 = 336.40818 = 336 m
BT = 250 / sin62 = 283.14251 = 283 m

b)

CosㄥTAB = (AT^2 + AB^2 - BT^2) / (2 AT AB)

CosㄥTAB = (336.40818^2 + 310^2 - 283.14251^2) / (2 * 336.40818 * 310)

CosㄥTAB = 0.61897

ㄥTAB = 51.7589 = 51.8°

c)

Let X be a point on AB such that OX 丄 AB ,

the required angle = ㄥTXO ,

The area of △TAB =

(1/2) AT * AB sinㄥTAB = (1/2)AB * TX

336.40818 * 310 * sin 51.7589 = 310 TX

TX = 264.22

sin ㄥTXO = TO/TX = 250 / 264.22

ㄥTXO = 71.117 = 71.1°

d)

No.

sin ㄥTPO = TO / TP , since the largest length of TP = TX ,

hence sin ㄥTPO = TO/TP =< TO/TX = 250/264.22,

ㄥTPO =< 71.1° , 78° is impossible.

2010-05-26 19:36:45 補充：
It should be the smallest length of TP = TX ,

hence sin ㄥTPO = TO/TP >= TO/TX = 250/264.22,

ㄥTPO =< 71.1° , 78° is impossible.

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