Sam and markus are standing on level ground 100m apart. A large tree is due north of markus and on a bearing of 65degrees from sam. The top of the tree appears at an angle of elevation of 25degrees to sam and 15degrees to markus. Find the height of the tree!

Well, it seems like the tree is playing a game of hide-and-seek with both Sam and Markus. Sneaky tree!

To find the height of the tree, we can use a bit of trigonometry. Let's call the height of the tree "h."

First, let's consider Sam's perspective. The angle of elevation from Sam to the top of the tree is 25 degrees. So, we can use the tangent function:

tan(25) = h / 100

Alright, now let's switch to Markus' point of view. The angle of elevation from Markus to the top of the tree is 15 degrees. Remember, the tree is due north of Markus, so we're dealing with a right-angled triangle. Again, we can use the tangent function:

tan(15) = h / x

Here, "x" represents the distance from Markus to the tree. To find "x," we can use some basic trigonometry. Since Sam and Markus are 100m apart, and the tree is on a bearing of 65 degrees from Sam, we can use the Law of Cosines:

x² = 100² + h² - 2 * 100 * h * cos(65)

Now we have two equations with two variables. We can solve them simultaneously to find the value of "h."

But hey, instead of doing all that math, let's see if the tree has any better suggestions. Hey, tree, how tall are you?

To find the height of the tree, we can use trigonometry and the information given in the problem. Let's break down the problem step by step:

Step 1: Draw a diagram

To begin, let's draw a diagram to represent the given information. We have Sam (S) and Markus (M) standing on level ground 100m apart. The large tree (T) is located north of Markus (M) on a bearing of 65 degrees from Sam (S). The angles of elevation from Sam (S) to the top of the tree and from Markus (M) to the top of the tree are 25 degrees and 15 degrees, respectively.

```
S T
/|\
/ | \
25° | 65°
/ | \
/____|____\
| M |
15°
```

Step 2: Identify the right triangle

From the diagram, we can see that we have two right triangles: one triangle with Sam (S), the top of the tree (T), and the ground point directly below the tree (G1), and another triangle with Markus (M), the top of the tree (T), and the ground point directly below the tree (G2).

Let's label the height of the tree as h and the distances from S to G1 and M to G2 as x and y, respectively.

```
S T
/|\
/ | \
25° | 65°
/ | \
/____|____\
| M |
15° G1 G2

x y
```

Step 3: Determine the lengths of the sides of the triangles

Using trigonometry, we can determine the lengths of the sides of the triangles. For the triangle with Sam (S), the top of the tree (T), and the ground point directly below the tree (G1):

Length(G1T)/Length(ST) = tan(angle of elevation from S to T)
Length(G1T)/h = tan(25°)
Length(G1T) = h * tan(25°)

For the triangle with Markus (M), the top of the tree (T), and the ground point directly below the tree (G2):

Length(G2T)/Length(MT) = tan(angle of elevation from M to T)
Length(G2T)/h = tan(15°)
Length(G2T) = h * tan(15°)

Step 4: Use the distance between S and M to find the height of the tree

We know that the distance between Sam (S) and Markus (M) is 100m. We can express this distance using the lengths of the sides of the triangles:

Length(ST) + Length(G1T) + Length(G2T) + Length(MT) = Length(SM)
100 + h * tan(25°) + h * tan(15°) + h = 100

Simplifying the equation:

h * tan(25°) + h * tan(15°) + h = 0
h * (tan(25°) + tan(15°) + 1) = 0
h = 100 / (tan(25°) + tan(15°) + 1)

Using a calculator:

h ≈ 100 / (0.4663 + 0.2679 + 1)
h ≈ 100 / 1.7342
h ≈ 57.64 meters

Therefore, the height of the tree is approximately 57.64 meters.

To find the height of the tree, we will use the concept of trigonometry and set up two right triangles.

Let's refer to the following points:
- Sam's position: S
- Markus's position: M
- Top of the tree: T

First, let's find the distances between the points:
- Distance between Sam and Markus (SM): 100m (given)
- Distance between Sam and the top of the tree (ST): unknown
- Distance between Markus and the top of the tree (MT): unknown

Next, let's calculate the heights of the top of the tree from both Sam and Markus's viewpoints.

For Sam's viewpoint triangle, we have:
- Angle of elevation from Sam's viewpoint (ST_angle): 25 degrees (given)
- Distance between Sam and the top of the tree (ST): unknown
- Height of the top of the tree from Sam's viewpoint (ST_height): unknown

Using trigonometry, we can set up the following equation:
tan(ST_angle) = ST_height / ST

For Markus's viewpoint triangle, we have:
- Angle of elevation from Markus's viewpoint (MT_angle): 15 degrees (given)
- Distance between Markus and the top of the tree (MT): unknown
- Height of the top of the tree from Markus's viewpoint (MT_height): unknown

Using trigonometry, we can set up the following equation:
tan(MT_angle) = MT_height / MT

Now, we need to find the values for ST and MT.

From the information given, we know that the top of the tree appears at a bearing of 65 degrees from Sam. Since the tree is due north of Markus, the bearing from Markus to the tree would be the complement of 65 degrees, which is 90 - 65 = 25 degrees. Therefore, the angles of elevation from both viewpoints are the same, but the distances differ.

We can set up the following equation to relate the distances:
SM = ST + MT

Using the given values and equations, now we can solve for the unknown heights.

1. Solve for ST:
Based on the equation: tan(ST_angle) = ST_height / ST, we substitute the known values:
tan(25 degrees) = ST_height / ST
Solve for ST_height:
ST_height = ST * tan(25 degrees)

2. Solve for MT:
Based on the equation: tan(MT_angle) = MT_height / MT, we substitute the known values:
tan(15 degrees) = MT_height / MT
Solve for MT_height:
MT_height = MT * tan(15 degrees)

3. Solve for ST and MT:
Since SM = ST + MT and SM is given as 100m, we can substitute the known values and the derived equations:
100 = ST + MT

4. Solve the system of equations:
Substitute the derived equations for ST_height and MT_height into the equation SM = ST + MT:
100 = ST * tan(25 degrees) + MT * tan(15 degrees)

Now we have two equations with two unknowns. We can solve this system either algebraically or using numerical methods such as a calculator or spreadsheet.

Once we have the values for ST and MT, we can choose either height value (ST_height or MT_height) to determine the height of the tree.

Remember to double-check your calculations and units to ensure accuracy.

9+6

Draw a sketch such that the tree (T) is due north of Markus (M), and Sam (S) is due (approx.) NW of M, such that angle STM is 65°, since the bearing of T at S is 65°.

We do not know the angles S (α) nor M (β), nor the distances ST (x1) nor MT(x2). However, we do know the distance SM = 100m.

see
http://img600.imageshack.us/img600/7954/1289559223.jpg

Let the height of the tree be h, express x1 and x2 in terms of h.

Use the sine rule to relate α, β and the known angle 65°.
This way, there will be 2 equations relating sin(α), sin(β) and h.

The third relation is given by the fact that α+β+65°=180° (angles of a triangle).

Solve by trial and error for α=80°, β=34° and h=29m approximately.

Also, for information (approx.):
x1=108.8m and x2=62.5m.

Redo and check my work.