The angle of depression of the top and bottom of a tower as seen from the top of a 100m high cliff are 30degree and 60degree respectively. find the height of the tower
To find the height of the tower, we can use trigonometry and the concept of similar triangles. Here's how we can approach the problem:
Step 1: Draw a diagram
Start by drawing a diagram representing the situation described in the problem. Label the height of the cliff as "h" and the height of the tower as "x". Note that the angle of depression is the angle formed by the line of sight from an observer to an object below the horizontal line.
Step 2: Identify the right triangles
In this problem, we have two right triangles. One is formed by the cliff, the tower, and the line of sight to the top of the tower. The other right triangle is formed by the cliff, the tower, and the line of sight to the bottom of the tower.
Step 3: Determine the trigonometric ratios
Using the given angle of depression and the right triangles, we can determine the trigonometric ratios involved. The tangent of an angle in a right triangle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
For the top angle of depression:
tan(30°) = h / x
For the bottom angle of depression:
tan(60°) = (h + 100) / x
Step 4: Solve the system of equations
We can solve the above system of equations by substituting the value of h from the first equation into the second equation:
tan(60°) = (tan(30°) + 100) / x
Now, we can solve for x:
x = (tan(30°) + 100) / tan(60°)
Step 5: Calculate the height of the tower
Substitute the value of x into the first equation to find the height of the tower:
h = x * tan(30°)
Simplify the expression and solve for h:
h = ((tan(30°) + 100) / tan(60°)) * tan(30°)
Using a scientific calculator or trigonometric tables, evaluate the trigonometric functions and calculate the value of h.
Note: Make sure to use the appropriate units for the height (e.g., if the output of h is in meters, then the height of the tower will also be in meters).
This step-by-step process will allow you to find the value of h, which represents the height of the tower.
I assume you made a sketch.
On mine I let the distance between tower and cliff be x m
then tan60° = 100/x
x = 100/tan60°
In my smaller right-angled triangle at the top of my diagram ,
let the difference between height of tower and height of cliff be h
then tan 30 = h/x
h = xtan30
= (100/tan60)tan30
then height of tower = 100 - h
I will let you do all that button-pushing,
let me know what you got