Interesting problem?

Question

http://img328.imageshack.us/img328/8479/untitledwd8.jpg
Please answer step by step

Because the problem has figure, I have to scan and upload to the third-party site.
http://img328.imageshack.us/img328/8479/untitledwd8.jpg

Answer 1

Let x be the distance from the base.
Let α be the angle of inclination to the bottom of the statue. θ is the angle shown in the picture.

Then tan(θ + α) = 92/x
and tan(α) = 46/x
so α = atn(46/x)
and θ + α = atn(92/x)
or θ = atn(92/x) - α = atn(92/x) - atn(46/x)

You need to take the derivative of θ with respect to x and then set that equal to 0 to find the maximum.
The derivative of atn(x) is 1/(x^2+1).

dθ/dx = {1/[(92/x)^2 + 1]}d/dx[92/x]
- {1/[(46/x)^2 + 1]}d/dx[46/x]

dθ/dx = {1/[(92/x)^2 + 1]}*[-92/x^2]
- {1/[(46/x)^2 + 1]}*[-46/x^2]

dθ/dx = -92/(x^2 + 92^2) + 46/(x^2 + 46^2)
-92/(x^2 + 92^2) + 46/(x^2 + 46^2) = 0
x^2 = 92^2 - 2*46^2
x^2 = 46^2 * [2^2 - 2]
x = 46√2

NOTE TO IRONDUKE:
The problem says that the height of the statue is 92 feet INCLUDING THE PEDESTAL, WHICH IS 46 FT. This means that the figure of liberty is 46 ft. (the distance spanned by θ.) So tan(θ + α) = 92/x, not 138/x

Answer 2

Let the angle between the ground and the top of the pedestal =x
Let your angle theta = y
Let the distance from the base to the viewinng point be x
Then we have tan x = 46/x, and
tan (x+y) = 138/x
tan(x+y) = (tan x+ tan y)/(1-tan x tan y) =138/x
(46/x + tan y)/(1- 46/x tan y) =138/x
[(46+xtany)/x]/[x-46tan y)/x] =138/x
(46 + xtan y)/(x-46tan y) = 138/x
Cross multiplying, we get:
46x + x^2tan y = 138x - 6348tan y
6348tan y +x^2tany = 138x -46x = 92x
(6348 + x^2)tan y = 92x
tan y = 92x/(6348+x^2)
y = arctan [(92x)/(6348 + x^2)]
dy/dx =1/[1 + (92x/(6348 + x^2)^2][(x^2+6384)92-92x(2x)]/(x^2 + 6348)^2
This boils down to:
dy/dx = [92(x^2 +6348)-184x^2]/(x^2 +6384) +92^2x^2)
Setting numerator to zero gives:
x^2=6348 = 46sqrt(3)

Answer 3

Dude, you lazy ***. making us do your homework for you isn't going to help. Did you scan that from your textbooks? You dumbass