From the roof and basement of a house 20m high,the angle of elevation of the top of a tower are 45 and 60 degres respectively. Find the height of the tower.
(Ans=81.97m)
There is no figure in this solution so please add a picture to understand clearly.
Well, well, well, we've got some angles and heights to tangle with! Let's do this!
First, let's imagine our tower standing tall, minding its own business. We have two angles of elevation - 45 degrees from the roof and 60 degrees from the basement. It's like the tower is getting checked out from both ends!
Now, let's take a closer look. The angle of elevation from the roof forms a right-angled triangle with the height of the tower. We know the opposite side (the height from the roof) is 20m, and the angle is 45 degrees.
Using some trigonometry, we can say that the height of the tower is equal to the opposite side divided by the tangent of the angle. So, the height of the tower from the roof = 20m / tan(45).
But wait, we're not done clowning around yet! We also have the angle of elevation from the basement, which forms another right-angled triangle with the height of the tower. This time, the opposite side (the height from the basement) is what we're looking for.
Using the same logic as before, we can say that the height of the tower from the basement = x / tan(60), where x is the height we're out to find.
Now, here's the plot twist! The height of the tower from the roof IS the same as the height from the basement. So, we can equate the two expressions and solve for x:
20m / tan(45) = x / tan(60).
Now, let's crunch some numbers! Plug these values into a calculator, and you'll find that x, the height of the tower, is approximately 81.97m.
There you have it, my friend! The height of the tower is approximately 81.97m. Time to raise a toast to the tower! Cheers!
To find the height of the tower, we can use trigonometry and the given angles of elevation. Let's break down the problem into steps:
Step 1: Understand the problem and organize the given information.
- We have a house with a roof and a basement, both 20m high.
- The angle of elevation from the roof to the top of the tower is 45 degrees.
- The angle of elevation from the basement to the top of the tower is 60 degrees.
- We need to find the height of the tower.
Step 2: Draw a diagram and label it.
Let's draw a diagram to visualize the situation described in the problem. The house will be represented as a vertical line, and the tower will be represented as another vertical line extending above the house. Label the known values on the diagram.
_______________________________ <- Top of the tower
| / | |
| / | |
| / |H |
| / |E |
| / |I |
| / |G |
|/ |H |
| |T |
| | |
| | |
| 20m | |
|___________| <- House
Step 3: Identify the right triangles involved.
From the diagram, we can see that two right triangles are formed. One triangle is formed by the roof, the top of the tower, and a vertical line from the roof to the ground. The other triangle is formed by the basement, the top of the tower, and a vertical line from the basement to the ground.
Step 4: Apply trigonometric ratios.
In the first triangle (roof triangle):
- We have the angle of elevation (45 degrees) and the opposite side (20m).
- We want to find the adjacent side, which represents the height of the tower above the roof.
In the second triangle (basement triangle):
- We have the angle of elevation (60 degrees) and the opposite side (20m).
- We want to find the adjacent side, which represents the height of the tower above the basement.
Using the tangent function, which relates the opposite and adjacent sides of a right triangle, we can write the equations:
For the roof triangle: tan(45) = height of tower above roof / 20m
For the basement triangle: tan(60) = height of tower above basement / 20m
Step 5: Solve the equations.
Let's solve the two equations simultaneously to find the height of the tower above the roof and the basement.
For the roof triangle: tan(45) = height of tower above roof / 20m
Rearranging the equation, we get: height of tower above roof = 20m * tan(45)
For the basement triangle: tan(60) = height of tower above basement / 20m
Rearranging the equation, we get: height of tower above basement = 20m * tan(60)
Using a scientific calculator, calculate the values for tan(45) and tan(60), and then substitute these values in the equations above.
The height of the tower above the roof is approximately:
height of tower above roof = 20m * 1 = 20m
The height of the tower above the basement is approximately:
height of tower above basement = 20m * sqrt(3) ≈ 34.64m
Step 6: Calculate the total height of the tower.
To find the total height of the tower, we need to add the heights above the roof and the basement:
Total height of tower = height above roof + height above basement
Total height of tower = 20m + 34.64m ≈ 54.64m
Therefore, the height of the tower is approximately 54.64m.-
h = height of tower
d = distance to tower
h / d = tan(60º)
(h - 20) / d = tan(45º)
h / (h - 20) = tan(60º) / tan(45º)
h = tan(60º) * (h - 20)
h [tan(60º) - 1] = 20 * tan(60º)