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 previous post: Wed., 9-7-16, 3:18 AM.
To answer these questions, we can use trigonometry, specifically the tangent function.
a. To find the distance from the church where the second elevation was taken, we need to determine how much closer the person moved to the church. We know that the angle of elevation increased by 20 degrees. Let's call the distance from the first point (where the 35-degree angle was measured) to the church "x" meters. Therefore, the distance from the second point (where the 55-degree angle was measured) to the church would be "x - 30" meters.
We can set up the following equation using the tangent function:
tan(35 degrees) = (height of the church spire) / x
Since we are looking for the value of "x - 30", we can use the same equation with the new elevation angle:
tan(55 degrees) = (height of the church spire) / (x - 30)
We can set up a ratio by dividing the second equation by the first:
tan(55 degrees) / tan(35 degrees) = (height of the church spire) / (x - 30) / (height of the church spire) / x
Now we can solve for "x - 30":
(x - 30) = x * (tan(55 degrees) / tan(35 degrees))
Simplifying further, we have:
x - 30 = x * tan(55 degrees) / tan(35 degrees)
Next, we isolate "x" by moving the "-30" term to the other side of the equation:
x - x * tan(55 degrees) / tan(35 degrees) = 30
Factoring out "x" on the left side:
x * (1 - tan(55 degrees) / tan(35 degrees)) = 30
Finally, we can solve for "x":
x = 30 / (1 - tan(55 degrees) / tan(35 degrees))
Calculating this value will give you the distance from the church where the second elevation was taken.
b. To calculate the height of the church spire, we can use the value of "x" we just found. Substituting this value back into the first equation:
tan(35 degrees) = (height of the church spire) / x
Now we can calculate the height of the church spire by multiplying "x" by the tangent of the 35-degree angle:
height of the church spire = x * tan(35 degrees)
By calculating this expression, you will find the height of the church spire.