the angle of elevation of a tower at place A south of it is 30degrees and at B west of A and a distance of 50 from it the angle of elevation is 18degrees, determined the height of the tower
Drawing a diagram in geometry is a must. All we need to do is to follow instructions step by step. If you refrain from drawing diagrams, you will be handicapped in almost ALL geometry problems.
This is how you could proceed:
"the angle of elevation of a tower at place A south of it is 30degrees"
"Let
a = distance from A to the base of the tower "
So draw any point in the middle of a piece of blank paper, call this point T (for tower).
Then draw a point about 1" below it and call this A (A is south of T, and by convention, south is below, north is above, east is to the right, and west is to the left).
" B west of A and a distance of 50 from it"
Draw a point B about an inch to the left (west) of A.
"b = distance from B to the base of the tower
h = height of the tower "
Label the distance from A to T as "a" and the distance from B to T as "b".
Your diagram is now complete.
The following lines are deduced from the diagram, without which it is almost impossible to do (unless the person has a mental diagram).
"Then we have
b^2 = a^2+50^2
h/b = tan18°
h/a = tan30° "
Jack, school work is meant to help you practise skills you have learned. Finding answers is doing yourself a disfavour, because you will be deprived of the exercise needed to master the skill.
It may seem hard work to try to understand, but understanding will help you solve other similar, related problems, as well as understand more difficult concepts you will learn later on. You will also be able to solve future problems related to your profession, work, everyday life, etc.
The satisfaction of solving problems by yourself is your reward for the hard work you put in today.
thanks dude..
To determine the height of the tower, we can use trigonometry and the concept of similar triangles.
Let's denote the height of the tower as h and the distance from point A to the tower as x.
From point A, the angle of elevation to the top of the tower is 30 degrees. This forms a right triangle. The opposite side is the height of the tower (h), and the distance from A to the tower (x) represents the adjacent side.
Using trigonometry, we can express this relationship using the tangent function:
tan(30°) = h / x
Next, let's consider the angle of elevation from point B. The angle of elevation to the top of the tower is 18 degrees. This also forms a right triangle, with the height of the tower (h) as the opposite side and the distance from B to the tower (50 + x) as the adjacent side.
Using trigonometry again, we can express this relationship using the tangent function:
tan(18°) = h / (50 + x)
Now we have two equations:
1. tan(30°) = h / x
2. tan(18°) = h / (50 + x)
We can solve these two equations simultaneously to find the height of the tower (h). Let's simplify and solve:
1. tan(30°) = h / x
√3/3 = h / x
h = (√3/3) * x
2. tan(18°) = h / (50 + x)
tan(18°) = (√3/3) * x / (50 + x)
Now, we can solve this equation for x:
tan(18°) = (√3/3) * x / (50 + x)
Cross-multiplying:
tan(18°) * (50 + x) = (√3/3) * x
Expanding:
50 * tan(18°) + tan(18°) * x = (√3/3) * x
Moving terms with x to one side:
tan(18°) * x - (√3/3) * x = -50 * tan(18°)
Factoring out x:
x * (tan(18°) - (√3/3)) = -50 * tan(18°)
Dividing both sides by (tan(18°) - (√3/3)):
x = (-50 * tan(18°)) / (tan(18°) - (√3/3))
Finally, substitute the value of x back into the equation for h:
h = (√3/3) * x
And you will have the height of the tower (h).
Draw a diagram. Let
a = distance from A to the base of the tower
b = distance from B to the base of the tower
h = height of the tower
Then we have
b^2 = a^2+50^2
h/b = tan18°
h/a = tan30°
Now you can find a,b and thus h.