a man measured the angle of elevation of the too of a tower to be 70°.when he walked 30m further,the angle of elevation of the top of the tower was 35°.find yhe diatance from the top of the tower to the second observation point.
Did you make a sketch?
Look at the triangle formed by the two observation points and the top of
the tower. We know all the angles, and we have a side, so we can use
the sine law.
Let the distance you want be x
x/sin110° = 30/sin35°
x = 30sin110/sin35 = appr 49.15 m
Check by doing it another way:
notice the triangle is also isosceles, so by the cosine law:
x^2 = 30^2 + 30^2 - 2(30)(30)cos110
= 900+900-1800cos110
= 2415.63...
x = 49.15 m, same as above
or, starting with the usual problem of finding the height (h) of the tower, you could do
h cot35° - h cot70° = 30
h = 30/(cot35° - cot70°) = 28.19
Now the desired distance (x) is
h/x = sin35°
x = h/sin35° = 49.15 as above
To solve this problem, we can use trigonometry, specifically the tangent function. Let's break down the steps:
1. We'll assume that the man is standing at point A when he measures the angle of elevation of the top of the tower to be 70°.
2. Let's call the distance from point A to the base of the tower "x".
3. We can then calculate the height of the tower using the tangent of the angle of elevation. The tangent of an angle can be found by dividing the opposite side by the adjacent side of a right triangle. In this case, the opposite side is the height of the tower and the adjacent side is the distance from point A to the tower base. So, the height of the tower is given by "x * tan(70°)".
4. Now, after walking 30m further, the man is at point B and measures the angle of elevation of the top of the tower as 35°. The distance from point A to point B is "x + 30".
5. We can again calculate the height of the tower from point B using the tangent function, similar to step 3. The height of the tower from point B is given by "(x + 30) * tan(35°)".
6. The difference between the two heights calculated in steps 3 and 5 gives us the actual height of the tower: "(x + 30) * tan(35°) - x * tan(70°)".
7. Since the actual height of the tower is the same in both scenarios, we can set the equation from step 6 equal to zero and solve for x.
- "(x + 30) * tan(35°) - x * tan(70°) = 0"
- Simplifying this equation will give us the value of x, which is the distance from the top of the tower to the second observation point.
So, to find the distance from the top of the tower to the second observation point, we need to solve the equation above.