Question 1186123
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Let points A,B,C be placed on the ground. 
A and B are the observation points.
C is the base of the tower.
D is the top of the tower.
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locate(1.1,-2.1,"A"),locate(-1.9,-2.1,"B"),locate(1.1,1.9,"C"),locate(3.1,-2.1,"B"),locate(5.1,-2.1,"A"),locate(6.14,-1.06,"C"),locate(6.1,1.9,"D"),
locate(1.2,0.08,x),locate(-0.84,-2.26,3),locate(-0.88,0.18,y),locate(4-0.5,-1.22+0.15,30^o),locate(6.26,1.2,h),locate(5.72,-1.5,60^o),

locate(3,-3,matrix(1,3,"angleCAD","=",60^o)),
locate(3,-3.5,matrix(1,3,"angleCBD","=",30^o)),

locate(-2,3,matrix(1,3,"Birds","Eye","View")),
locate(4,3,matrix(1,2,3D,"View")),
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)
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x = distance from A to C
y = distance from B to C
h = height of the tower = distance from C to D
Each distance is in kilometers and each is positive.


Focus on right triangle CAD.
tan(angle) = opposite/adjacent
tan(angle CAD) = CD/CA
tan(60) = h/x
h = x*tan(60)
h = x*sqrt(3)


Now focus on right triangle CBD.
tan(angle) = opposite/adjacent
tan(angle CBD) = CD/CB
tan(30) = h/y
h = y*tan(30)
x*sqrt(3) = y*tan(30) ........... plugged in h = x*sqrt(3)
x*sqrt(3) = y*(1/sqrt(3))
y = x*sqrt(3)*sqrt(3)
y = 3x


Focus on right triangle ABC that is entirely on the ground.
The horizontal leg (that runs east/west) is AB = 3 km.
The leg running north/south is AC = x and the hypotenuse is BC = y = 3x


Use the Pythagorean Theorem for right triangle ABC.
(AB)^2 + (AC)^2 = (BC)^2
(3)^2 + (x)^2 = (3x)^2
9 + x^2 = 9x^2
9x^2-x^2 = 9
8x^2 = 9
x^2 = 9/8
x = sqrt(9/8)
Since x > 0 we ignore the minus of the plus/minus.


Then,
h = x*sqrt(3)
h = sqrt(9/8)*sqrt(3)
h = sqrt((9/8)*3)
h = sqrt(27/8)
h = sqrt(27)/sqrt(8)
h = (3*sqrt(3))/(2*sqrt(2))
h = (3*sqrt(3)*sqrt(2))/(2*sqrt(2)*sqrt(2)) .... rationalizing denominator
h = (3*sqrt(3*2))/(2*2)
h = <font color=red>(3/4)*sqrt(6) km</font> is the exact answer
h = <font color=red>1.837117 km approximately</font>


The approximate answer varies depending how you round. 


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Another way to solve:


Focus on triangle CAD
tan(60) = h/x
x = h/tan(60)
x = h/sqrt(3)


Focus on triangle CBD
tan(30) = h/y
y = h/tan(30)
y = h*sqrt(3)


The last triangle to focus on is ABC.
It has legs AB = 3 and AC = x. The hypotenuse is BC = y.
Use the Pythagorean Theorem
(AB)^2 + (AC)^2 = (BC)^2
3^2 + x^2 = y^2
3^2 + (h/sqrt(3))^2 = (h*sqrt(3))^2
9 + (h^2)/3 = h^2*3
27 + h^2 = 9h^2
9h^2-h^2 = 27
8h^2 = 27
h^2 = 27/8
h = sqrt(27/8)


The steps from here are the same as the previous section.
You should arrive at
h = <font color=red>(3/4)*sqrt(6) km</font> = <font color=red>1.837117 km approximately</font> 


As of 2025, the world's tallest building is the Burj Khalifa at around 828 meters tall. 
The answer we got is over 1000 meters taller, so tower CD is definitely breaking records.
I don't know how realistic it is for a tower to be this tall. 
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