The angle of elevation of the top of a flagpole from a point on the level ground is 30degrees. From another point on the ground 20m, nearer the pole the angle of elevation is 60degrees. Calculate the height of the pole?
1) y2+6y+9-4y2 its standard form.
its degree.
2) expand using identity (5b-1/2)2(square)
3) find the product of 3/2xyz(4/9x2(square)y2-2xy)
if the nearer point is at distance x from a pole of height h,
h/x = tan 60°
h/(x+20) = tan 30°
substituting the first into the second,
h = (h/tan60° + 20)tan30°
h = (h/√3 + 20)(1/√3)
h = 10(3+√3) = 47.32m
To solve this problem, we can use trigonometry. Let's consider the side opposite to the angle of elevation as the height of the pole.
Let h be the height of the pole.
From the first point on the ground, the angle of elevation is 30 degrees. This forms a right-angled triangle with the height of the pole as the opposite side. We can use the tangent function to find the height:
tan(30°) = h / x (where x is the distance from the first point to the pole)
From the second point on the ground, the angle of elevation is 60 degrees. This also forms a right-angled triangle with the height of the pole as the opposite side. Similarly, we can use the tangent function to find the height:
tan(60°) = h / (x - 20) (where x - 20 is the distance from the second point to the pole)
To find the height of the pole, we can set these two equations equal to each other:
h / x = tan(30°)
h / (x - 20) = tan(60°)
Now, let's solve these equations step by step:
Step 1: Rewrite the equations using the values of the tangent of 30° and 60°:
(1)
h / x = √3 / 3
(2)
h / (x - 20) = √3
Step 2: Cross-multiply in equation (1) to eliminate the fractions:
h = (x * √3) / 3
Step 3: Substitute the value of h from equation (2) into equation (1):
(x * √3) / 3 = √3 * (x - 20)
Step 4: Multiply both sides by 3 to eliminate the fractions:
x * √3 = 3 * √3 * (x - 20)
Step 5: Simplify and distribute:
x * √3 = 3√3x - 60√3
Step 6: Move all terms with x to one side and all constants to the other side:
x√3 - 3√3x = -60√3
Step 7: Factor out x from the left side:
x(√3 - 3√3) = -60√3
Step 8: Simplify:
(x - 3x)√3 = -60√3
Step 9: Combine like terms:
-2x√3 = -60√3
Step 10: Cancel out √3 on both sides:
-2x = -60
Step 11: Solve for x:
x = -60 / -2
x = 30
Step 12: Substitute the value of x back into equation (1) to find h:
h = (30 * √3) / 3
h = 10√3
Therefore, the height of the pole is 10√3 meters.
To calculate the height of the pole, we can use trigonometric ratios.
Let's denote the height of the pole as 'h'.
From the information given in the problem, we have two right-angled triangles. The first triangle is formed by the flagpole, the point where the person is standing, and the top of the flagpole. The second triangle is formed by the flagpole, the point where the other person is standing (20m closer), and the top of the flagpole.
To solve this problem, we can break it down into two steps:
Step 1: Find the length of the adjacent side of the first triangle.
- In the first triangle, the angle of elevation is 30 degrees. Therefore, the length of the adjacent side to this angle is equal to the distance from the person to the flagpole.
- Let's denote this distance as 'x'.
Step 2: Find the length of the adjacent side of the second triangle.
- In the second triangle, the angle of elevation is 60 degrees. Therefore, the length of the adjacent side to this angle is equal to the distance from the other person (20 m closer) to the flagpole.
- Let's denote this distance as 'x - 20'.
Now, we can use trigonometric ratios to find the height of the pole.
In the first triangle:
tan(30 degrees) = h / x
In the second triangle:
tan(60 degrees) = h / (x - 20)
We can solve these equations simultaneously to find the value of 'h'.
Let's substitute the values into the equations:
tan(30 degrees) = h / x
tan(60 degrees) = h / (x - 20)
By substituting the values of the tangents of these angles (approximately 0.5773 and 1.732), we can solve the equations by cross-multiplication.