from a certain spot, the top of a triangle had an angle of elevation of 40 degree. move 15meter in a straight line towards the flag pole.Now to the top has an angle of elevation of 60degree.Find the height of the flagpole and it's distance from the second point.
If the distance on the ground is x, and the height of the pole is h,
h cot40° - h cot60° = 15
x/h = cot60°
To find the height of the flagpole and its distance from the second point, we can use trigonometry. Let's break down the problem step by step:
Step 1: Draw a diagram.
Draw a triangle to represent the situation described. Label the points and measurements as follows:
- Let A be the initial spot.
- Let B be the second point after moving 15 meters towards the flagpole.
- Let C be the top of the flagpole.
- Label the height of the flagpole as h, and the distance from B to C as x.
Now, based on the given information, we have an angle of elevation of 40 degrees at point A and 60 degrees at point B.
Step 2: Identify the right triangles.
From the diagram, we can identify two right triangles: triangle ABC and triangle ADB.
Step 3: Solve for the height of the flagpole (h).
In triangle ABC, we can use the tangent function to find the height (h):
tan(40 degrees) = h / x
Solve for h:
h = x * tan(40 degrees) ------ Equation 1
Step 4: Solve for the distance from B to C (x).
In triangle ADB, we can use the tangent function to find x:
tan(60 degrees) = h / (x + 15)
Solve for x:
x = (h / tan(60 degrees)) - 15 ------ Equation 2
Step 5: Combine equations 1 and 2 to find the values of h and x.
Substitute equation 1 into equation 2:
x = [(x * tan(40 degrees)) / tan(60 degrees)] - 15
Multiply both sides by tan(60 degrees):
x * tan(60 degrees) = x * tan(40 degrees) - 15 * tan(60 degrees)
Simplify:
x * tan(60 degrees) - x * tan(40 degrees) = -15 * tan(60 degrees)
Factor out x:
x * (tan(60 degrees) - tan(40 degrees)) = -15 * tan(60 degrees)
Divide both sides by (tan(60 degrees) - tan(40 degrees)):
x = [-15 * tan(60 degrees)] / [tan(60 degrees) - tan(40 degrees)]
Now, you can use a scientific calculator or trigonometric tables to determine the values of tan(40 degrees) and tan(60 degrees). Plug in those values and calculate x to find the distance from B to C.
Once you have the value of x, substitute it into equation 1 to find the height of the flagpole (h).
So, by following these steps and using the appropriate trigonometric calculations, you can find the height of the flagpole and its distance from the second point after moving 15 meters.