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 trigonometry and create a diagram to illustrate the situation.
Let's denote the distance from the first point to the base of the spire as "x" and the height of the spire as "h".
a. To find the distance from the second elevation point to the base of the spire, we need to consider the difference in the two angles of elevation.
From the first point:
Tan(35°) = h / x
From the second point (30 meters closer):
Tan(35° + 20°) = h / (x - 30)
We can solve these two equations simultaneously to find "x - 30".
b. To calculate the height of the church spire, we need to solve for "h".
From the first equation, we can rearrange it to:
h = x * Tan(35°)
Using the value of "x" from part a, we can substitute it into this equation to find the height.
Let's proceed with the calculations.
a. Solving for the distance from the second elevation point to the base of the spire:
Tan(35°) = h / x
Therefore, x = h / Tan(35°)
Similarly,
Tan(35° + 20°) = h / (x - 30)
Substituting the value of "x" from the previous equation:
Tan(35° + 20°) = h / (h / Tan(35°) - 30)
Simplifying further, we have:
Tan(55°) = Tan(35°) * (Tan(35°) - 30)
Now, we solve this equation for h.
b. Solving for the height of the church spire:
h = x * Tan(35°) [from the first equation]
Since we have calculated the value of "x" in part a, we can substitute it into this equation to find the height.
With these calculations, we can determine both the distance from the second elevation point to the base of the spire and the height of the church spire.