On the horizontal plane there is a vertical tower is a vertical tower and the shadow of its is found to be 40 m longer.when sun's altitude is 30 degree than what it is 60 degree.find the height of tower.
draw a diagram. Let the height of the tower be h, and the length of shadow for a 60deg sun be x. Then we have
h/x = √3
h/(x+40) = 1/√3
√3 x = (x+40)/√3
3x = x+40
x = 20
h = 20√3 = 34.64m
To find the height of the tower, we can use trigonometry and the concept of similar triangles.
Let's consider the situation when the sun's altitude is 30 degrees. In this case, we have a right triangle formed by the tower, its shadow, and the line from the top of the tower to the top of the shadow (let's call this line "x").
Using the tangent function, we can set up the following equation:
tan(30 degrees) = height of the tower / x
Simplifying, we have:
1/sqrt(3) = height of the tower / x
Now, we are given that the shadow of the tower is 40 meters longer than its height. So, we can set up another equation:
x + 40 = height of the tower
Now, let's consider the case when the sun's altitude is 60 degrees. Again, we have a right triangle formed by the tower, its shadow, and the line from the top of the tower to the top of the shadow (let's call this line "y").
Using the tangent function again, we can set up the following equation:
tan(60 degrees) = height of the tower / y
Simplifying, we have:
sqrt(3) = height of the tower / y
Now, we know that the height of the tower is the same in both cases, so we can equate the values of x and y:
x = y
Substituting the expressions for x and y into the equation above, we get:
x + 40 = sqrt(3) * (x + 40)
Now, we can solve this equation to find the value of x and then calculate the height of the tower.
Here's how you can solve the equation:
1. Distribute the sqrt(3) term on the right side of the equation:
x + 40 = sqrt(3) * x + 40 * sqrt(3)
x + 40 = sqrt(3) * x + 40 * sqrt(3)
2. Subtract x from both sides of the equation:
40 = sqrt(3) * x + 40 * sqrt(3) - x
3. Simplify the equation by combining like terms:
40 = (sqrt(3) - 1) * x + 40 * sqrt(3)
4. Subtract 40 * sqrt(3) from both sides of the equation:
40 - 40 * sqrt(3) = (sqrt(3) - 1) * x
5. Divide both sides of the equation by (sqrt(3) - 1):
(40 - 40 * sqrt(3)) / (sqrt(3) - 1) = x
Now, calculate the value of x using a calculator or computer software:
x ≈ 94.87 meters
Finally, substitute this value of x into the equation x + 40 = height of the tower:
94.87 + 40 = height of the tower
Height of the tower ≈ 134.87 meters
Therefore, the height of the tower is approximately 134.87 meters.