a tower stands on the top of a cliff. at a distance of 55m from the foot of the cliff, the angles of elevation of the top of the tower as well the cliff are 60° and 45° respectively. find the height of the tower.
as usual, draw a diagram. You will see that the distances to top and bottom of tower are
top/55 = tan 60°
bottom/55 = tan 45°
Now just evaluate top and bottom, and you can subtract to get the height of the tower.
You should get 55(√3 - 1)
Well, well, well, looks like we've got a towering question here, quite literally! So let's dive into it.
First things first, let's break it down. We have a tower standing on top of a cliff, and we've got a distance of 55m from the foot of the cliff. The angle of elevation of the top of the tower is 60°, and the angle of elevation of the cliff is 45°. We need to find the height of the tower.
Now, here comes the fun part. Since both angles of elevation form a right triangle, we can use some good old trigonometry to solve this.
Let's call the height of the tower "h". Now, we've got a right triangle, with the opposite side being "h", the adjacent side being 55m (since it's the distance from the foot of the cliff), and we're given the angle of elevation of the top of the tower as 60°.
Using the tangent function, we can write:
tan(60°) = h/55
Now, it's time to get our hands dirty with some math. Using your favorite calculator, plug in tan(60°), and multiply it by 55:
h = tan(60°) * 55
And the result is... *drumroll please* ... h ≈ 95.10m!
So, there you have it! The height of the tower is approximately 95.10 meters. Hope that brings you closer to the heights of satisfaction!
To find the height of the tower, we can use trigonometric ratios. Let's consider the right-angled triangle formed by the tower, the cliff, and the distance from the foot of the cliff to the tower.
Let:
- The height of the tower be represented by 'h'.
- The distance from the foot of the cliff to the tower be represented by 'x'.
From the problem, we know that:
- The angle of elevation of the top of the tower from a point on the ground is 60°.
- The angle of elevation of the cliff from the same point on the ground is 45°.
- The distance between the foot of the cliff and the tower is 55m.
Using the tangent function, we can relate the height 'h' and the distance 'x' as follows:
tan(60°) = h / x (1)
tan(45°) = h / (x + 55) (2)
Now, let's solve these equations step-by-step:
Step 1: Solve equation (1) for 'x':
tan(60°) = h / x
√3 = h / x (as tan(60°) = √3)
Step 2: Solve equation (2) for 'x':
tan(45°) = h / (x + 55)
1 = h / (x + 55) (as tan(45°) = 1)
Step 3: Substitute the value of h from equation (1) into equation (2):
1 = (√3)(x) / (x + 55)
Step 4: Cross-multiply and solve the equation:
x + 55 = √3x
55 = √3x - x
55 = (√3 - 1)x
x = 55 / (√3 - 1)
Step 5: Calculate the value of 'h' using equation (1):
√3 = h / x
h = √3 * (55 / (√3 - 1))
Step 6: Simplify the expression for 'h':
h = 55√3 / (√3 - 1)
Thus, the height of the tower is approximately 81.94 m.
To find the height of the tower, we can use trigonometry and the given angles of elevation.
Let's start by labeling the diagram:
- Let "A" be the foot of the cliff.
- Let "B" be the top of the tower.
- Let "C" be the point on the ground directly below the top of the tower.
We need to find the height of the tower, which is the length of line segment BC.
We are given the following information:
- AB = 55m (the distance from the foot of the cliff to the tower)
- Angle CAB = 60° (angle of elevation from point A to point B)
- Angle ACB = 45° (angle of elevation from point A to point C)
Now, let's use trigonometry to solve for BC:
First, let's find the length of AC. We can use the tangent function:
tan(45°) = AC / AB
1 = AC / 55m
AC = 55m (since tan(45°) = 1)
Next, let's find the length of BC. We can use the tangent function again:
tan(60°) = BC / AC
√3 = BC / 55m
BC = √3 * 55m
BC ≈ 95.26m
Therefore, the height of the tower, BC, is approximately 95.26m.