# Find the height of the tree - understanding the task

• Vital
In summary, the conversation is about a problem involving the use of the Law of Sines to find the height of a tree. The problem is initially misunderstood, but the correct assumptions and understanding of the task are clarified through a discussion of a diagram. It is suggested to use the Law of Sines with the 6 degree angle and ∠CAQ in order to find the tree height directly.
Vital

## Homework Statement

Hello!
Please, help me to understand the task - I seem to fail to understand what goes where, and hence cannot proceed to solving the exercise. Please, take a look at the task, and then my questions. The task is on using the law of sines. Before trying to solve it I need to understand the exercise itself.

## Homework Equations

Along a long, straight stretch of mountain road with a 7% grade, you see a tall tree standing

perfectly plumb alongside the road. From a point 500 feet downhill from the tree, the angle

of inclination from the road to the top of the tree is 6. Use the Law of Sines to find the

height of the tree. (Hint: First show that the tree makes a 94 angle with the road.)

## The Attempt at a Solution

I would like to draw what's going on with this road. I am attaching a horrific picture (horrific because I draw it on the trackpad, and as it is a pure square it was not easy to draw anything), but at least it gives some idea.

Do I assume correctly that:
(1) 500 feet is the length of the horizontal leg of a right triangle, CB on the picture;
(2) then the trunk of a tree forms a vertical leg of a right triangle, AB on the picture;
(3) A is the top of the tree, so angle of inclination 6 is angle formed by ACQ (C in the middle);
(4) I need to find the angle QCB (with C in the middle).

Are these correct assumptions and a correct understanding of the task?
Thank you!

#### Attachments

• Screen Shot 2017-05-05 at 19.13.45.png
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Vital said:

## Homework Statement

Hello!
Please, help me to understand the task - I seem to fail to understand what goes where, and hence cannot proceed to solving the exercise. Please, take a look at the task, and then my questions. The task is on using the law of sines. Before trying to solve it I need to understand the exercise itself.

## Homework Equations

Along a long, straight stretch of mountain road with a 7% grade, you see a tall tree standing

perfectly plumb alongside the road. From a point 500 feet downhill from the tree, the angle

of inclination from the road to the top of the tree is 6. Use the Law of Sines to find the

height of the tree. (Hint: First show that the tree makes a 94 angle with the road.)

## The Attempt at a Solution

I would like to draw what's going on with this road. I am attaching a horrific picture (horrific because I draw it on the trackpad, and as it is a pure square it was not easy to draw anything), but at least it gives some idea.

Do I assume correctly that:
(1) 500 feet is the length of the horizontal leg of a right triangle, CB on the picture;
(2) then the trunk of a tree forms a vertical leg of a right triangle, AB on the picture;
(3) A is the top of the tree, so angle of inclination 6 is angle formed by ACQ (C in the middle);
(4) I need to find the angle QCB (with C in the middle).

Are these correct assumptions and a correct understanding of the task?
Thank you!
You have drawn the tree tilted over so it forms a 90 degree angle with the road. The problem says that the tree is "plumb", which means that it is vertical. Can you make a more detailed drawing showing the tree being vertical?

berkeman said:
You have drawn the tree tilted over so it forms a 90 degree angle with the road. The problem says that the tree is "plumb", which means that it is vertical. Can you make a more detailed drawing showing the tree being vertical?
Actually, no, the tree on my picture doesn't form a 90 degree with the road. The tree is AB. Road is CQ. And CA is the view point from the road to the top of the tree.
Is this incorrect?

Vital said:
...
Do I assume correctly that:
(1) 500 feet is the length of the horizontal leg of a right triangle, CB on the picture;
(2) then the trunk of a tree forms a vertical leg of a right triangle, AB on the picture;
(3) A is the top of the tree, so angle of inclination 6 is angle formed by ACQ (C in the middle);
(4) I need to find the angle QCB (with C in the middle).

Are these correct assumptions and a correct understanding of the task?
Thank you!
For (1) :
It seems more likely that the 500 feet is along the road. That's CQ in your sketch.​

For (2) :
The tree trunk is AQ .​

(3) is fine.

For (4) :
If you are going to use the Law of Sines to get the tree height directly, you should consider using ∠CAQ (It's opposite the 500 foot side) and use the 6° angle, ∠ACQ.
.

berkeman
SammyS said:
For (1) :
It seems more likely that the 500 feet is along the road. That's CQ in your sketch.​

For (2) :
The tree trunk is AQ .​

(3) is fine.

For (4) :
If you are going to use the Law of Sines to get the tree height directly, you should consider using ∠CAQ (It's opposite the 500 foot side) and use the 6° angle, ∠ACQ.
.
I see. Thank you. I will try it.

I am stuck. Don't see a picture.

Vital said:
I am stuck. Don't see a picture.
It's the picture you provided.

AQ is the tree.

CQ is the road (length: 500 ft.).

SammyS said:
It's the picture you provided.
View attachment 199293

AQ is the tree.

CQ is the road (length: 500 ft.).
Yes, yes. By saying that I don't see a picture, I was talking about a big picture, metaphorically :) Not about the one I draw :)
Do I understand correctly that if the road CQ has 7% grade, then ∠CQA cannot be 90°?

Vital said:
Yes, yes. By saying that I don't see a picture, I was talking about a big picture, metaphorically :) Not about the one I draw :)
Do I understand correctly that if the road CQ has 7% grade, then ∠CQA cannot be 90°?

Well, you can see that for yourself: ∠CBA = 90°, so what does that tell you?

## 1. How do I determine the height of a tree?

To determine the height of a tree, you will need to use a measuring tool, such as a measuring tape or ruler, and follow a few simple steps. First, stand at a distance from the tree where you can see the entire height of the tree. Then, measure the distance from the base of the tree to the top of the tree. This will give you the height of the tree.

## 2. Do I need any special equipment to find the height of a tree?

The only equipment you will need to find the height of a tree is a measuring tool, such as a measuring tape or ruler. However, if the tree is very tall, you may need a ladder or someone to help you measure from a higher point.

## 3. Why is it important to accurately measure the height of a tree?

Accurately measuring the height of a tree can provide valuable information about the tree's age, growth rate, and overall health. It can also help with tree management and planning for things like pruning or tree removal.

## 4. Are there different methods for finding the height of a tree?

Yes, there are several methods for finding the height of a tree, including the stick method, the pencil method, and the tangent method. Each method involves using basic trigonometry to calculate the height of the tree.

## 5. Can I estimate the height of a tree without measuring?

While it is not as accurate as measuring, you can estimate the height of a tree using your own height as a reference. First, stand at a distance from the tree where you can see the entire height. Then, measure your own height and compare it to the height of the tree in the same units (such as feet or meters). This will give you a rough estimate of the tree's height.

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