Two boys are on opposite sides of a tower.They sight the top of the tower at 33 degree's and 24 degree angles of elevation respectively.If the height of the tower is 100 m, find the distance between the two boys.
To solve this problem, we can use trigonometry and the concept of similar triangles.
Let's denote the distance between one of the boys and the tower as "x" and the distance between the other boy and the tower as "y". We need to find the value of x + y.
First, let's consider the triangle formed by one of the boys, the top of the tower, and the base of the tower. The angle of elevation is 33 degrees, and we are given the height of the tower, which is 100 m. This gives us a right-angled triangle with the height (opposite side) as 100 m and the base (adjacent side) as x.
We can use the tangent function to find x:
tan(33 degrees) = opposite / adjacent
tan(33 degrees) = 100 / x
Next, let's consider the triangle formed by the other boy, the top of the tower, and the base of the tower. The angle of elevation is 24 degrees, and we are again given the height of the tower, which is 100 m. This gives us another right-angled triangle with the height (opposite side) as 100 m and the base (adjacent side) as y.
Using the same tangent function:
tan(24 degrees) = opposite / adjacent
tan(24 degrees) = 100 / y
Now, we have two equations:
1) tan(33 degrees) = 100 / x
2) tan(24 degrees) = 100 / y
We can rearrange equation 1 to solve for x:
x = 100 / tan(33 degrees)
Similarly, we can rearrange equation 2 to solve for y:
y = 100 / tan(24 degrees)
Now, we can substitute these values of x and y into the expression x + y to find the distance between the two boys:
x + y = (100 / tan(33 degrees)) + (100 / tan(24 degrees))
Calculating this expression will give us the answer to the question.
Note: Make sure your calculator is set to degrees mode when calculating the tangent of angles.