If the angle of a h meters long tower is in a straight line between two observers A and B angle of elevation of the top of the tower from A and B are alpha and beta respectively and AB=D meters find h/d
say d = x + y
tan alpha = h/x
tan beta = h/y
so x = h/tan alpha
and y = h/tan beta
d = x + y = h (1/tan alpha + 1/tan beta)
or
d = h (cot alpha + cot beta)
h/d = 1/(cot alpha + cot beata)
Tnx damon
To find the value of h/d, we need to apply some trigonometric principles. Let's break down the problem step by step:
1. Let's draw a diagram to visualize the problem.
```
C
/|
/ |
h/ |b
/ |
/ |
A _______ |
\ a \ |
\ \|
\________ B
d
```
2. From the diagram, we can see that AB represents the distance between observers A and B, and h represents the height of the tower. We are given the angle of elevation of the top of the tower from both A and B (alpha and beta, respectively).
3. From observer A's perspective, the angle of elevation to the top of the tower is α. This means that we can form a right-angled triangle with sides h, a, and b (where a is the distance from the base of the tower to observer A and b is the distance from the top of the tower to observer A).
4. Similarly, from observer B's perspective, the angle of elevation to the top of the tower is β. This means we can form another right-angled triangle with sides h, c, and d (where c is the distance from the base of the tower to observer B and d is the distance from the top of the tower to observer B).
5. We need to find h/d, so we need to find the values of h and d using the given information. To do this, we'll use the tangent trigonometric function.
6. In triangle ABC, we have:
tan(α) = h / a (Equation 1)
=> h = tan(α) * a (Equation 2)
7. In triangle BCD, we have:
tan(β) = h / c (Equation 3)
=> h = tan(β) * c (Equation 4)
8. We can equate Equations 2 and 4 since h is the same in both cases:
tan(α) * a = tan(β) * c
9. Since AB = D = a + c, we can rewrite the equation as:
tan(α) * (D - c) = tan(β) * c
10. Let's solve the equation for c (distance from the base of the tower to observer B):
tan(α) * D - tan(α) * c = tan(β) * c
tan(α) * D = (tan(β) + tan(α)) * c
c = (tan(α) * D) / (tan(β) + tan(α))
11. Now that we have the value of c, we can substitute it into Equation 4 to find h:
h = tan(β) * c
12. Finally, we can find the value of h/d:
h/d = tan(β) * c / d
By following these steps and substituting the values of α, β, and D, you can find the ratio h/d.