A man standing 18m due west from the base of a vertical tower observes that the angle of elevation of the is 60⁰. He then walks a distance of 25m in a direction of N ∅⁰ W (where ∅ is acute) and observes that the angle of elevation is now 40⁰. Calculate, correct to one decimal place
A) The height of the tower
B) The distance of the man from the tower in the second position of observation
C) The value of ∅.
To solve this problem, we can use trigonometry and the concept of right triangles.
Let's label the given information:
1. The distance from the man to the base of the tower is 18m (adjacent side).
2. The angle of elevation from the man to the top of the tower is 60 degrees.
3. The man then walks a distance of 25m in a direction of N ∅⁰ W (where ∅ is acute).
4. The angle of elevation from the man's second position to the top of the tower is 40 degrees.
To calculate the height of the tower (A):
We can use the tangent function since we know the length of the adjacent side (18m) and the angle of elevation (60 degrees).
tan(60 degrees) = opposite/adjacent
tan(60 degrees) = height/18m
Rearranging the equation:
height = tan(60 degrees) * 18m
Using a calculator to find the tangent of 60 degrees:
height = 1.732 * 18m = 30.984m
So the height of the tower is approximately 31.0m (rounded to one decimal place).
To calculate the distance of the man from the tower in the second position of observation (B):
We can use the tangent function again, this time with the angle of elevation of 40 degrees.
tan(40 degrees) = opposite/adjacent
tan(40 degrees) = height/distance
We can rearrange the equation to solve for the distance:
distance = height / tan(40 degrees)
Substituting the height we found in the previous calculation (30.984m):
distance = 30.984m / tan(40 degrees)
Using a calculator to find the tangent of 40 degrees:
distance = 30.984 / 0.839
distance = 36.92m
So the distance of the man from the tower in the second position of observation is approximately 36.9m (rounded to one decimal place).
To calculate the value of ∅ (angle measures between North and West):
To find the angle ∅, we need to consider the change in position from the man's initial position to his second position.
If the change in position is in the direction of N ∅⁰ W, then we can subtract the angles from a full 90 degrees (90 degrees - N ∅⁰ W) to determine ∅.
90 degrees - N ∅⁰ W = ∅ degrees
Since ∅ is acute, meaning it is less than 90 degrees, we can determine ∅ by finding the supplement of N ∅⁰ W.
Supplement of an angle is 180 degrees minus the angle itself.
Supplement of N ∅⁰ W = 180 degrees - ∅ degrees
From the problem, we know that the angle of elevation at the second position is 40 degrees, so:
180 degrees - 40 degrees = 140 degrees
Therefore, the value of ∅ is 140 degrees.
Let's solve this step by step.
Step 1: Determine the height of the tower.
In the given triangle, the opposite side to the angle of elevation can be assumed as the height of the tower. Let's represent it as 'h'.
Using the trigonometric function tangent, we can determine the height:
tan(60°) = h/18
h = 18 * tan(60°)
Using a calculator, we can find the height to be approximately 31.2 meters.
So, the height of the tower is 31.2 meters.
Step 2: Determine the distance of the man from the tower in the second position of observation.
Let's represent this distance as 'x'.
In the right triangle formed by the man's second position of observation, the adjacent side to the angle of elevation is the distance from the tower. The opposite side to the angle of elevation can be assumed as the height of the tower, which we already found in Step 1 (h = 31.2 m).
Using the trigonometric function tangent, we can determine the distance from the tower:
tan(40°) = 31.2/x
Rearranging the equation, we get:
x = 31.2/tan(40°)
Using a calculator, we can find the distance to be approximately 38.3 meters.
So, the distance of the man from the tower in the second position of observation is approximately 38.3 meters.
Step 3: Determine the value of ∅.
We are given that the man walks a distance of 25m in a direction of N ∅⁰ W. Let's represent the angle ∅ as 'a'. The compass direction N ∅⁰ W indicates the angle relative to North.
In the right triangle formed by the man's starting position, the hypotenuse is the distance walked by the man (25m), and the opposite side to angle a is the distance parallel to the North direction. The adjacent side to angle a is the distance parallel to the West direction (18m).
Using the trigonometric function tangent, we can determine the value of a:
tan(a) = distance parallel to North / distance parallel to West
tan(a) = 25 / 18
Using a calculator, we can find the value of a to be approximately 51.3 degrees.
So, the value of ∅ is approximately 51.3 degrees.
If the height of the tower is h, then
h/18 = tan60°
from the new position, at a distance x,
h/x = tan40°
If we label a diagram with
T = base of tower
A = initial position
B = final position of man
If T is at (0,0) then B is at (h,k) where
h = -18-25sin∅
k = 25 cos∅
and h^2 + k^2 = x^2