two points A and B 80 ft. apart lie on the same side of a tower and in a horizontal line through its foot. if the angle of elevation of the top of the tower at A is 21 degree and B is 46 degree, find the height of the tower.
To find the height of the tower, we can use the concept of trigonometry. Let's call the height of the tower "h".
Step 1: Find the distance from point A to the tower:
tan(21 degrees) = h / distance from A to the tower
tan(21 degrees) = h / (distance from B to the tower + 80 ft)
Step 2: Find the distance from point B to the tower:
tan(46 degrees) = h / distance from B to the tower
tan(46 degrees) = h / (distance from A to the tower - 80 ft)
Step 3: Solve the equations simultaneously to find the height of the tower (h).
From Step 1:
tan(21 degrees) = h / (distance from B to the tower + 80 ft)
Rearranging the equation, we have:
(distance from B to the tower + 80 ft) = h / tan(21 degrees) ---- (equation 1)
From Step 2:
tan(46 degrees) = h / (distance from A to the tower)
Rearranging the equation, we have:
(distance from A to the tower) = h / tan(46 degrees) ---- (equation 2)
Step 4: Substitute equation 1 and equation 2 into each other to solve for h.
(distance from B to the tower + 80 ft) = h / tan(21 degrees)
(distance from B to the tower + 80 ft) = (h / tan(46 degrees)) / tan(21 degrees)
(distance from B to the tower + 80 ft) = h / (tan(46 degrees) * tan(21 degrees))
Substitute (distance from A to the tower) with h / tan(46 degrees):
(distance from B to the tower + 80 ft) = (h / tan(46 degrees) / tan(21 degrees))
Rearrange to isolate h on one side:
h = (distance from B to the tower + 80 ft) * (tan(46 degrees) * tan(21 degrees))
Now, you can substitute the values of (distance from B to the tower) and solve for h.
To find the height of the tower, we can use trigonometry and create a right triangle.
Let's label the points: A and B are the two points on the ground, and T is the top of the tower.
Looking at the triangle, we have the following information:
- Side AB is 80 ft long, and it represents the distance between points A and B.
- We have the angle of elevation at point A, which is 21 degrees, and the angle of elevation at point B, which is 46 degrees.
- We need to find the height of the tower, which is represented by side AT.
To solve this problem, we can use the tangent function. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side in a right triangle.
Let's calculate the height of the tower using the tangent function in two steps:
Step 1: Calculate the length of side BT.
In triangle ABT, we have the angle of elevation at point B, which is 46 degrees. The tangent of this angle is equal to the ratio of the height of the tower (AT) to the distance from point B to the tower (BT). We can write this as:
tan(46°) = AT / BT
Step 2: Calculate the height of the tower (AT).
In triangle AET, we have the angle of elevation at point A, which is 21 degrees. The tangent of this angle is equal to the ratio of the height of the tower (AT) to the distance from point A to the tower (ET). We can write this as:
tan(21°) = AT / ET
Since ET + BT = AB, we can rewrite the second equation as:
tan(21°) = AT / (AB - BT)
Now we have two equations with two unknowns (AT and BT). We can solve them simultaneously to find the values of AT and BT.
Once we have the value of AT, we will have the height of the tower.
Let P be the foot of the tower, and h be the height of the tower
let AP = x , then BP = 80-x
I see two right-angled triangles
tan21 = h/x ---> h = xtan21
tan46 = h/(80-x) --> h = (80-x)tan46
xtan21 = (80-x)tan46
xtan21 = 80tan46 - xtan46
xtan21 + xtan46 = 80tan46
x(tan21 + tan46) = 80tan46
x = 80tan46/(tan21 + tan46) ----- I have not yet touched a calculator
you do the button pushing to get x
then plug x into
h = xtan21
Two points A and B 80 ft apart lie on the same side of a tower
and in a horizontal line through its foot. If the angle of
elevation of the top of the tower at A is 21o and at B is 46o,
find the height of the tower