The angle of elevation of the top of a tower is 27 degrees from point A due east of it. The angle of elevation of the top of the tower is 11 degrees from another point B due south of the tower. Find the height of the tower if the distance AB is 40m.
mmmh, I just noticed that Steve answered this question for you earlier.
Why are you reposting it ?
Steve answer is not clear..
To find the height of the tower, we can use trigonometric ratios and the given angles.
First, let's draw a diagram to visualize the situation:
```
Tower
/\
/ \
/ \ A
/ \
/ \
/ |
B |_________|
```
Now, we can break down the problem into two right triangles. Triangle OAB formed by points A, B, and the base of the tower is a right triangle. Triangle OBC formed by points B, the top of the tower, and the base of the tower is also a right triangle.
Let's assign variables:
- Let h be the height of the tower.
- Let x be the horizontal distance from the tower's base to point A.
In triangle OAB:
- The angle of elevation from point A gives us the opposite side as h.
- The distance AB gives us the adjacent side as x.
In triangle OBC:
- The angle of elevation from point B gives us the opposite side as h.
- The distance AB gives us the adjacent side as 40 - x (as AB is the sum of x and the remaining distance from B to the base of the tower).
Now, we can set up the trigonometric ratios:
In triangle OAB:
- tan(27 degrees) = h / x (1)
In triangle OBC:
- tan(11 degrees) = h / (40 - x) (2)
We have two equations with two unknowns (h and x). We can solve this system of equations to find the values.
First, let's isolate h in equation (1):
h = x * tan(27 degrees) (3)
Next, substitute equation (3) into equation (2):
tan(11 degrees) = (x * tan(27 degrees)) / (40 - x)
Now, we can solve this equation for x. Multiply both sides by (40 - x):
tan(11 degrees) * (40 - x) = x * tan(27 degrees)
Expand and rearrange:
440 * tan(11 degrees) - tan(11 degrees) * x = tan(27 degrees) * x
Move the x term to one side:
tan(11 degrees) * x + tan(27 degrees) * x = 440 * tan(11 degrees)
Factor out x:
x * (tan(11 degrees) + tan(27 degrees)) = 440 * tan(11 degrees)
Divide both sides by (tan(11 degrees) + tan(27 degrees)):
x = (440 * tan(11 degrees)) / (tan(11 degrees) + tan(27 degrees))
Now, substitute this value of x into equation (3) to find h:
h = x * tan(27 degrees)
Finally, calculate the values of x and h using the given angles and distances.