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.
why keep posting the same problem over and over? You know, it just may be that no one is awake in the middle of the night.
Surely by now you could have drawn a diagram and seen that if the 2nd elevation was taken at distance x, and the height is h, then
h cot35° - h cot55° = 30
x = h cot55°
Shut it
To solve this problem, we can use trigonometric ratios, specifically the tangent (tan) function, which relates the angle of elevation to the distance and height of an object.
Let's go step by step to find the answers to both parts of the question:
a. To find how far from the church the second elevation was taken, we need to determine the distance between the two points where the angles of elevation were measured.
Let's assume the initial distance from the first point to the church is 'x' meters. When walking closer to the church by 30 meters, the new distance becomes 'x - 30' meters.
We can set up a trigonometric equation using the tangent function:
tan(35 degrees) = height of the spire / x
tan(35 + 20 degrees) = height of the spire / (x - 30)
Now, we need to solve these two equations simultaneously to find the value of 'x', which represents the distance from the church where the second elevation was taken.
First, let's find the tangent values:
tan(35 degrees) ≈ 0.7002
tan(55 degrees) ≈ 1.4281
Now we have our equations:
0.7002 = height of the spire / x
1.4281 = height of the spire / (x - 30)
To eliminate the height of the spire, we can divide the second equation by the first:
1.4281 / 0.7002 = (height of the spire / (x - 30)) / (height of the spire / x)
Simplifying:
2.0398 = (x / (x - 30))
Cross multiplying:
2.0398(x - 30) = x
Expanding and rearranging:
2.0398x - 61.194 = x
Subtracting 'x' from both sides:
1.0398x = 61.194
Dividing by 1.0398:
x ≈ 58.91
Therefore, the second elevation was taken approximately 58.91 meters from the church.
b. To calculate the height of the church spire, we can use either of the initial equations. Let's use the first equation:
0.7002 = height of the spire / x
Substituting the value of 'x':
0.7002 = height of the spire / 58.91
Now, we can solve for the height of the spire:
height of the spire ≈ 0.7002 * 58.91
height of the spire ≈ 41.26 meters
Therefore, the height of the church spire is approximately 41.26 meters.