from a particular point the angle of elevation of the top of a church spire is 35 degrees. walking 30 metres closer to the church the angle of elevation of the top of the spire increases by 20 degrees.

a. how far from the church was the second elevation taken?
b. calculate the height of the church spire.

To solve this problem, we can use trigonometric ratios, specifically the tangent function. Let's start by breaking down the problem step by step.

a. To find how far from the church the second elevation was taken, we need to determine the distance between the two elevations. Let's assume the distance from the initial point to the church is "x" meters.

When the angle of elevation is 35 degrees, we have a right triangle formed between the initial point, the top of the spire, and the church. The opposite side is the height of the spire, and the adjacent side is the distance from the initial point to the church (x meters).

Using the tangent function, we can establish the equation:
tan(35°) = height of spire / x

Simplifying this equation, we have:
height of spire = x * tan(35°)

Now, let's consider the second elevation. When the angle of elevation increases by 20 degrees, we form another right triangle with the same height of the spire but a shorter distance between the second elevation and the church. Let's call this distance "d" meters.

Using the tangent function again, we can establish the equation:
tan(35° + 20°) = height of spire / (x + 30)

Simplifying this equation, we have:
height of spire = (x + 30) * tan(55°)

Since the height of the spire remains constant between the two elevations, we can set the two equations equal to each other:
x * tan(35°) = (x + 30) * tan(55°)

Now, we can solve this equation to find the value of x, which represents the distance from the church where the second elevation was taken.

b. To calculate the height of the church spire, we can use either of the two equations mentioned earlier:

- height of spire = x * tan(35°) (using the first elevation distance)
- height of spire = (x + 30) * tan(55°) (using the second elevation distance)

By substituting the respective values of x in either equation, we can determine the height of the church spire.