The angle of elevation of the top of a tower as observed from a point on the ground is alpha and on moving a meters towards the tower, the angle of elevation is beta .prove that the height of the tower is a (tan alpha*tan beta)/(tan beta - tan alpha)
make a sketch, label the height as h
distance from last position to the tower as x
tan (alpha) = h/(x+a)
h = (x+a)tan alpha
in the same way
h = xtan beta
(x+a)tan alpha = xtan beta
x tan alpha - xtan beta = - a tan alpha
x(tan alpah - tan beta) = -a tan alpha
x = -a tan alpha/(tan alpha - tan beta)
x = a tan alpha /(tan beta - tan alpha)
h = xtan beta
= a tan alpha * tan beta/(tan beta - tan alpha)
as requested
To prove the given expression for the height of the tower, we can use the concept of trigonometry and create a right triangle with the given information.
Let's assume that the height of the tower is represented by 'h'.
In the right triangle, the vertical length (opposite side) is 'h', the horizontal length (adjacent side) is 'a', and the angle of elevation at the original position is 'α'. When we move 'a' meters towards the tower, the angle of elevation becomes 'β'.
Using the definition of tangent, we can write the following equations:
tan(α) = h/a ---(1)
tan(β) = h/(a+h) ---(2)
Now, we need to find a way to eliminate 'h' from these equations. To do that, we can rearrange equation (2) as follows:
tan(β) = h/a + h/(a+h)
tan(β) = (h(a+h) + ah) / a(a+h)
tan(β) = (h(a+h) + ah) / (a^2 + ah)
tan(β) = (a^2 + 2ah + ah) / (a^2 + ah)
tan(β) = (a^2 + 3ah) / (a^2 + ah)
Cross-multiplying, we get:
(a^2 + ah) * tan(β) = (a^2 + 3ah)
Expanding the equation, we have:
a^2 * tan(β) + ah * tan(β) = a^2 + 3ah
Rearranging and factoring out 'h', we obtain:
h * (tan(β) - tan(α)) = a^2 - a^2 * tan(β)
Dividing both sides by (tan(β) - tan(α)), we get:
h = (a^2 - a^2 * tan(β))/(tan(β) - tan(α))
h = a^2 * (1 - tan(β))/(tan(β) - tan(α))
Finally, dividing both the numerator and denominator by a^2, we get:
h = a * (1 - tan(β))/(tan(β) - tan(α))
h = a * (tan(β) - 1)/(tan(β) - tan(α))
Therefore, the height of the tower is indeed given by the expression: h = a * (tan(β) - 1)/(tan(β) - tan(α)).
To prove the given formula for the height of the tower, we can use the concept of trigonometry and create a mathematical equation based on the information provided.
Let's consider the situation described:
- Angle of elevation from the ground to the top of the tower, observed at a point, is α.
- On moving a meters closer to the tower, the angle of elevation becomes β.
Now, we need to find an expression that represents the height of the tower using the given information.
Step 1: Set up a diagram
To visualize the situation more clearly, draw a vertical line to represent the tower. At the bottom of the tower, draw a line that represents the ground. Mark the point of observation with a dot and label it as point A.
Tower
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| A
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Ground | /
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Step 2: Identify the relevant trigonometric ratios
Notice that we can form two right triangles with the tower, point A, and the two angles of elevation α and β.
In the first triangle (larger one), the sides are:
- Opposite side: Height of the tower (let's call it h)
- Adjacent side: a (since the person moves closer to the tower)
- Hypotenuse: unknown (let's call it x)
In the second triangle (smaller one), the sides are:
- Opposite side: Height of the tower minus a (h - a)
- Adjacent side: a
- Hypotenuse: unknown (x)
Now, we can establish the following trigonometric ratios for each triangle:
For the larger triangle:
- tan(α) = h / x ---(1)
For the smaller triangle:
- tan(β) = (h - a) / x ---(2)
Step 3: Eliminate x
From equations (1) and (2), we can observe that both equations involve the same value of x. So, we can solve for x in both equations and then equate them to eliminate x.
From equation (1): x = h / tan(α)
From equation (2): x = (h - a) / tan(β)
Setting these two expressions for x equal to each other, we get:
h / tan(α) = (h - a) / tan(β)
Step 4: Solve for h
To find the height of the tower (h), rearrange the equation:
h * tan(β) = (h - a) * tan(α)
Expanding and simplifying:
h * tan(β) = h * tan(α) - a * tan(α)
Now, isolate h on one side of the equation:
h * (tan(β) - tan(α)) = a * tan(α)
Finally, divide both sides by (tan(β) - tan(α)):
h = (a * tan(α)) / (tan(β) - tan(α))
Therefore, we have proved that the height of the tower is given by the formula:
h = (a * tan(α)) / (tan(β) - tan(α))
Note: It's important to remember to always double-check your calculations and assumptions when solving mathematical equations.