SOLUTION: Two points in line with a tower and in the same horizontal plane with its base are 180 feet apart. From the point nearer the tower the angle of elevation of the top of the tower is

Algebra ->  Trigonometry-basics -> SOLUTION: Two points in line with a tower and in the same horizontal plane with its base are 180 feet apart. From the point nearer the tower the angle of elevation of the top of the tower is      Log On


   

Question 762977: Two points in line with a tower and in the same horizontal plane with its base are 180 feet apart. From the point nearer the tower the angle of elevation of the top of the tower is A, from the other point the angle of elevation is B. If sinA=3/5 and cosB=12/13, What is the height of the tower?
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
Assume both points are on the same side of the tower.
Let
h+=+height of the tower
x+=+distance from base of tower to A
180+%2B+x+=+distance from base of tower to+B
sinA+=+3%2F5
cosA+=+4%2F5
tanA+=+sinA+%2F+cosA+=+%283%2F5%29+%2F+%284%2F5%29+=+3%2F4
cosB+=+12%2F13
sinB+=+5%2F13
tanB+=+sinB%2FcosB+=+%285%2F13%29+%2F+%2812%2F13%29+=+5%2F12
tanA+=+3%2F4+=+h%2Fx
tanB+=+5%2F12+=+h%2F%28180+%2B+x%29
3%2F4+=+h%2Fx
x+=+4h%2F3
5%2F12+=+h%2F%28180+%2B+x%29
5%28180+%2B+x%29+=+12h
900+%2B+5x+=+12h
5x+=+12h+-+900
x+=+%2812h+-+900%29%2F5
Set the two equations equal and solve for h.
+4h%2F3+=+%2812h+-+900%29%2F5
5%284h%29+=+3%2812h+-+900%29
20h+=+36h+-+2700
-16h+=+-2700
h+=-2700%2F-16feet
h+=2700%2F16feet
h+=168.75feet

2.The radius of a circle is 100 feet. Find the perimeter and the area of
The area of any polygon is given by:
A=%28n%2F4%29s%5E2%2Acot%28pi%2Fn%29 or A=%28s%5E2n%29%2F4tan%28%28pi%2Fn%29%29
where,
s is the length of any side
n is the number of sides
PI is approximately 3.142
or
Area of the polygon A=n*area of triangle AOB.....n is the number of sides
area of triangle AOB=(n/2)*OB*OA*sin(360/n)
OB*OA=r*r=r^2
since given radius and the number of sides , we will use this formula:
so, Area of the polygon A=n%2A%28n%2F2%29%2Ar%5E2%2Asin%28360%2Fn%29

a) a regular circumscribed pentagon
n=5 and r=100ft
A=5%2A%285%2F2%29%2A%28100ft%29%5E2%2Asin%28360%2F5%29
A=%2825%2F2%29%2A%28100ft%29%5E2%2Asin%2872%29
A=12.5%2A%28100ft%29%5E2%2A0.95106
A=118882ft%5E2+
to find side s we can use formula
A=s%5E2n%2F%284tan%28pi%2Fn%29%29
118882+ft%5E2+=s%5E2%2A5%2F%284tan%283.14%2F5%29%29
s%5E2=%284tan%283.14%2F5%29%2A118882+ft%5E2%29%2F5
s%5E2=%284tan%280.628%29%2A118882+ft%5E2%29%2F5
s%5E2=%284%280.726056%29%2A118882+ft%5E2%29%2F5
s%5E2=%28345259.96ft%5E2%29%2F5
s%5E2=69052+ft%5E2+
s=262.78ft
the perimeter P=5%2A262.78ft=1313.9ft

b) a regular circumscribed decagon.

do all same just use n=10 and sin%2836%29}
n=10 and r=100ft
A=10%2A%2810%2F2%29%2A%28100ft%29%5E2%2Asin%28360%2F10%29

A=50%2A%28100ft%29%5E2%2Asin%2836%29
A=50%2A%28100ft%29%5E2%2A0.587785
A=293893ft%5E2+
293893ft%5E2+=s%5E2%2A10%2F%284tan%283.14%2F10%29%29
s%5E2=%284tan%283.14%2F10%29%2A293893ft%5E2%29%2F10
s%5E2=%284tan%280.314%29%2A293893ft%5E2%29%2F10
s%5E2=%284%280.324744%29%2A293893ft%5E2%29%2F10
s%5E2=38176ft%5E2+

s=195.39ft
the perimeter P=10%2A195.39ft=1953.9ft