a man measured the angle of elevation of the top of a tower to be 70 degree. When he walked 300 m. further, the angle of elevation of the top was 35 degree. What is the height of the tower?
make a diagram
label the tower AB , where A is the top of the tower.
label the man's first position C, his second position D
so that DC = 300
Look at triangle ADC,
angle D = 35°
angle ACD = 110° , so then
angle DAC = 35°
so the triangle is isosceles (lucky) and AC = 300
Then in the right-angled triangle ACB ,
sin70 = AB/300
AB = 300sin70 = 281.9 m
(had the triangle not been isosceles, I would have used the Sine Law to find AC )
A BOY OBSERVED THAT THE ANGLE OF ELEVATION OF THE TOP OF A TOWER IS 32 Degree HE THEN WALKS 8m Towards The Tower And THEN DISCOVER THAT THE OF Angle Of Elevation Is 43degree .Find The Height Of The Tower
To find the height of the tower, we can use the tangent function. Let's consider two right triangles formed: one where the man is standing at the original position, and another where the man is standing 300 m further.
In the first triangle, let's label the height of the tower as "h1" and the distance between the man and the base of the tower as "d1". In the second triangle, let's label the height of the tower as "h2" and the distance between the man and the base of the tower as "d2".
From triangle 1, we have the equation: tan(70) = h1/d1
From triangle 2, we have the equation: tan(35) = h2/d2
Since the man walks 300 m further, we know that d2 = d1 + 300.
Now, we need to eliminate d1 and d2 from the equations to solve for the height of the tower.
Rearranging the first equation, we have h1 = d1 * tan(70).
Substituting d2 = d1 + 300 in the second equation, we have tan(35) = h2/(d1 + 300).
Now, we can set up an equation to solve for h2.
h2 = (d1 + 300) * tan(35).
Since h1 = h2, we can equate the two equations:
d1 * tan(70) = (d1 + 300) * tan(35).
Now, we can solve for d1:
d1 * tan(70) = d1 * tan(35) + 300 * tan(35)
d1 * (tan(70) - tan(35)) = 300 * tan(35)
d1 = (300 * tan(35)) / (tan(70) - tan(35))
Finally, we can substitute the value of d1 back into the first equation to find the height of the tower:
h1 = d1 * tan(70)
To find the height of the tower, we can use trigonometry. Let's denote the height of the tower as 'h'.
We have two right triangles formed by the observer, the tower, and the ground. The first triangle is formed when the observer is at a certain point, and the second triangle is formed when the observer walks 300 meters further.
In the first triangle:
- The angle of elevation is 70 degrees.
- The distance between the observer and the tower is the height of the tower, h.
- The distance between the observer and the base of the tower is unknown. Let's call it 'x'.
In the second triangle:
- The angle of elevation is 35 degrees.
- The distance between the observer (who has moved 300 meters further) and the tower is still the height of the tower, h.
- The distance between the observer (who has moved 300 meters further) and the base of the tower is x + 300 meters.
Now, we can set up the following equations:
In the first triangle, tan(70°) = h / x.
In the second triangle, tan(35°) = h / (x + 300).
We can solve this system of equations to find the value of h:
tan(70°) = h / x
=> x = h / tan(70°) ... (Equation 1)
tan(35°) = h / (x + 300)
=> h / (x + 300) = tan(35°)
=> h = (x + 300) * tan(35°) ... (Equation 2)
Now, we can substitute Equation 1 into Equation 2 and solve for h:
(x + 300) * tan(35°) = h / tan(70°)
Multiplying both sides by tan(70°):
(x + 300) * tan(35°) * tan(70°) = h
Substituting tan(70°) = 2 * tan(35°):
(x + 300) * (2 * tan(35°) * tan(35°)) = h
(x + 300) * (2 * tan^2(35°)) = h
Using trigonometric identity tan^2(35°) = (1 - cos(2 * 35°)) / (1 + cos(2 * 35°)):
(x + 300) * (2 * (1 - cos(70°)) / (1 + cos(70°))) = h
Finally, we can calculate the value of h using this equation.
Note: It's crucial to convert the trigonometric functions' angles into radians if your calculator's mode is set to radians.