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.
See your previous post.
To find the answers, we can use trigonometry and set up some right-angled triangles.
a. To determine how far from the church the second elevation was taken, we need to calculate the horizontal distance. Let's denote the first position as point A and the second position as point B.
From point A, where the angle of elevation is 35 degrees, draw a line representing the spire, and label the top of the spire as point C.
From point A, draw a horizontal line to intersect with the vertical line from point C, and label this point of intersection as point D.
Now, consider the triangle ACD. The angle at point D is a right angle, and we know that the angle of elevation at point A is 35 degrees. Therefore, the angle at point C is 90 - 35 = 55 degrees.
Since angles in a triangle add up to 180 degrees, the angle at point A is also 180 - 90 - 55 = 35 degrees.
Next, consider triangle ABD. We are given that the angle of elevation at point B is 35 + 20 = 55 degrees.
We can now set up the following equation using the tangent function:
tan(55) = AC / AD.
We know that AD = 30 meters (as the person walked 30 meters closer to the church). Therefore, we can solve for AC.
AC = AD * tan(55) = 30 * tan(55) meters.
So, the second elevation was taken approximately 30 * tan(55) meters away from the church.
b. To calculate the height of the church spire, let's consider the triangle ABC.
We know that AB = 30 * tan(55) meters (from part a).
We need to find the height of the spire, BC.
Using the tangent function again, we can set up the following equation:
tan(35) = BC / AB.
We know AB from part a. Therefore, we can solve for BC.
BC = AB * tan(35) = (30 * tan(55)) * tan(35) meters.
So, the height of the church spire is approximately (30 * tan(55)) * tan(35) meters.