1) The angle of elevation to the bottom of a transmission tower on a hill from an observer standing 1.8 km away from the base of the hill is 5 degrees. The angle of elevation to the top of the tower from the tower is 6.8 degrees. If the distance from the observer to the base is 1.8 km, find the height of the tower to the nearest metre.
Tan5 = Y1/X = Y1/1.8
Y1 = 1.8*Tan5 = 0.157 km = Ht. of hill.
Tan6.8 = (Y1+Y2)/X = (Y1+Y2)/1.8
Y1+Y2 = 1.8*Tan6.8 = 0.215 km
Y2 = 0.215-Y1 = 0.215-0.157 = 0.058 km. = 58 m. = Ht. of tower.
To find the height of the tower, we can use trigonometry.
Let's consider the angle of elevation to the bottom of the transmission tower from the observer. We can create a right triangle with the tower's height (h), the distance from the observer to the base of the hill (1.8 km), and the distance from the observer to the bottom of the tower (d).
Using the tangent function, we have:
tan(5 degrees) = h / d
Rearranging the equation, we get:
h = d * tan(5 degrees)
Now, let's consider the angle of elevation to the top of the tower from the tower. Again, we can create a right triangle, this time with the height of the tower (h), the distance from the observer to the top of the tower (1.8 km + d), and the distance from the tower to the top of the tower (d).
Using the tangent function, we have:
tan(6.8 degrees) = h / (1.8 km + d)
Rearranging the equation, we get:
h = (1.8 km + d) * tan(6.8 degrees)
Since the height of the tower should be the same in both equations, we can set them equal to each other:
d * tan(5 degrees) = (1.8 km + d) * tan(6.8 degrees)
Now, we can solve this equation to find the value of d.
To find the height of the tower, we can use trigonometry. Let's denote the height of the tower as 'h'.
Given that the angle of elevation to the bottom of the transmission tower is 5 degrees, we can form a right triangle. The side adjacent to the angle of elevation is the distance from the observer to the base of the hill, which is 1.8 km.
Using trigonometry, we can say that:
tan(5 degrees) = h / 1.8 km
Next, let's consider the angle of elevation to the top of the tower from the tower itself, which is 6.8 degrees. Again, we can form another right triangle. The side adjacent to this angle is the height of the tower, 'h'.
Using trigonometry, we can say that:
tan(6.8 degrees) = h / d
where 'd' is the distance from the observer to the top of the tower. We can find 'd' by using the Pythagorean theorem. The distance from the observer to the base is 1.8 km, and the distance from the top of the tower to the base (which is 'd') can be represented as:
d = √((1.8 km)^2 + h^2)
Now, we have two equations:
Equation 1: tan(5 degrees) = h / 1.8 km
Equation 2: tan(6.8 degrees) = h / (√((1.8 km)^2 + h^2))
To solve these equations simultaneously, we can rearrange Equation 1 to solve for 'h', and then substitute that value into Equation 2:
h = 1.8 km * tan(5 degrees)
Substituting this value of 'h' into Equation 2, we have:
tan(6.8 degrees) = (1.8 km * tan(5 degrees)) / (√((1.8 km)^2 + (1.8 km * tan(5 degrees))^2))
Now, we can solve this equation to find the value of 'h'.