A TREE AND A FLAGPOLE ARE IN THE SAME GROUND.A BIRD ON TOP OF THE TREE OBSERVES THE TOP AND BOTTOM OF THE FLAGPOLE BELOW IT AT ANGLES OF 45 AND 60 DEGREES RESPECTIVELY. IF THE TREE IS 10.65m HIGH WHAT IS THE HEIGHT OF THE FLAGPOLE?
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Prove the answer
To solve this problem, we can use the concept of trigonometry. Let's break down the given information:
1. The bird on top of the tree observes the top and bottom of the flagpole.
2. The angle at the top of the flagpole (observed by the bird) is 45 degrees.
3. The angle at the bottom of the flagpole (also observed by the bird) is 60 degrees.
4. The height of the tree is given as 10.65m.
To find the height of the flagpole, we can set up a right triangle. The tree's height forms the vertical side, the distance from the bird to the bottom of the flagpole forms the base, and the distance from the bird to the top of the flagpole forms the hypotenuse.
Let's denote the height of the flagpole as x. Now we can use the trigonometric ratios:
1. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side.
tan(45 degrees) = height of the flagpole / distance to the top of the flagpole
tan(45 degrees) = x / distance to the top of the flagpole
2. The tangent of another angle is equal to the ratio of the opposite side to the adjacent side.
tan(60 degrees) = height of the flagpole / distance to the bottom of the flagpole
tan(60 degrees) = x / distance to the bottom of the flagpole
We know the distance between the bird and the bottom of the flagpole is the same as the distance between the bird and the top of the flagpole because they are in the same ground.
To find the distance to the bottom of the flagpole, we can use the Pythagorean theorem.
c^2 = a^2 + b^2
In this case, a is the distance from the bird to the top of the flagpole, b is the height of the tree, and c is the distance to the bottom of the flagpole.
Now let's calculate the distances:
a = x
b = 10.65m
Using the Pythagorean theorem, we can solve for c:
c^2 = a^2 + b^2
c^2 = x^2 + (10.65m)^2
c = √(x^2 + 113.5225m^2)
Now substitute these values into the equations using the trigonometric ratios:
tan(45 degrees) = x / c
tan(60 degrees) = x / c
By rearranging the equations, we can solve for x:
x = tan(45 degrees) * c
x = tan(60 degrees) * c
Finally, substitute the value of c:
x = tan(45 degrees) * √(x^2 + 113.5225m^2)
x = tan(60 degrees) * √(x^2 + 113.5225m^2)
At this point, you can use an algebraic solver or numerical approximation techniques to solve for x.