This is the question:

Two spruce trees are 100m apart. From the point on the ground halfway between the trees the angles of elevation are 21 degrees and 39 degrees. Determine the distance, to the nearest metre, between the tops of the two trees. Show your work.

find the heights of the two trees

... 50m tan(21º) and 50m tan(39º)

the difference in the heights is the vertical distance between the tops

the horizontal distance is 100m

(total distance)^2 = (vertical distance)^2 + (horizontal distance)^2

make your sketch , to get 2 right-angled triangles, both with a base of 50

From midpoint of base,
distance to top of taller tree ---- x
distance to top of smaller tree --- y
cos39 = 50/x ----> x = 50/cos39
cos21 = 50/y ---> y = 50/cos21

D^2 = x^2 + y^2 - 2xy cos120
D = ...

To solve this problem, we can use trigonometry and the concept of similar triangles.

Let's start by drawing a diagram to visualize the situation. We have two spruce trees, which we can represent as points A and B, as well as a point on the ground halfway between the trees, which we can represent as point C. We also have angles of elevation from point C to the tops of the trees, which are 21 degrees and 39 degrees. Lastly, we need to find the distance between the tops of the trees, which we can represent as x.

The diagram would look something like this:

A----D----B
| /
| /
| /
| /
| /
| /
| /
|/
C

Now, let's label the different lengths in the diagram. Let the height of tree A be h1, the height of tree B be h2, and the distance between tree A and C be d1.

Using trigonometry, we can establish the following relationships:

tan(21°) = h1 / d1
tan(39°) = h2 / (100 - d1)

To eliminate the variables h1 and h2, we can express them in terms of x:

h1 = x - d1
h2 = x + (100 - d1)

Now, let's substitute these expressions into the equations:

tan(21°) = (x - d1) / d1
tan(39°) = (x + (100 - d1)) / (100 - d1)

Now we have a system of two equations with two variables (x and d1). We can solve this system using substitution or elimination:

From the first equation, we can rewrite it as:
d1 = (x - (x / tan(21°))) / (1 / tan(21°))
Simplifying this expression gives us:
d1 = x * (1 / tan(21°)) / (1 - 1/tan(21°))

We can substitute this expression for d1 into the second equation:

tan(39°) = (x + (100 - (x * (1 / tan(21°)) / (1 - 1/tan(21°))))) / (100 - (x * (1 / tan(21°)) / (1 - 1/tan(21°))))

Now we have a single variable equation that we can solve for x. Once we find x, we can calculate the distance between the tops of the trees by substituting x back into one of the previous expressions (either h1 or h2).

Finally, we can round the distance between the tops of the trees to the nearest meter based on the given instructions.

To solve this problem, we can use trigonometry and the concept of similar triangles.

Let's break down the problem step by step:

Step 1: Draw a diagram
Start by drawing a diagram to visualize the situation. Draw two spruce trees 100 meters apart. Label the midpoint between the trees as point M. Then, draw two lines from points M to the tops of each tree, forming two right triangles.

Step 2: Identify the relationships
In our diagram, we have two right triangles with different angles of elevation. These angles are 21 degrees and 39 degrees. We know that angles of elevation are measured from the horizontal line (parallel to the ground).

Step 3: Identify the known and unknown values
We know the distance between the trees is 100 meters. We need to find the distance between the tops of the trees. Let's label this unknown distance as x.

Step 4: Set up the proportions
Since the two triangles share the same base (the line connecting the trees), they are similar triangles. This means that ratios of corresponding sides are equal.

For the first triangle, the opposite side is the height of the first tree, h1. The adjacent side is the distance from the midpoint M to the first tree, which is 50 meters.

For the second triangle, the opposite side is the height of the second tree, h2, and the adjacent side is the distance from the midpoint M to the second tree (also 50 meters).

So, we can set up two proportions:

tan(21 degrees) = h1 / 50
tan(39 degrees) = h2 / 50

Step 5: Solve the proportions for the heights
Using a scientific calculator, we can calculate the two heights (h1 and h2) using the given angles of elevation and the known adjacent side (50 meters).

h1 = 50 * tan(21 degrees)
h2 = 50 * tan(39 degrees)

Step 6: Find the distance between the tops of the trees
To find the distance between the tops of the trees, we need to add the heights of the two trees. So,

x = h1 + h2

Step 7: Calculate the final result
Substitute the values of h1 and h2 calculated in Step 5 into Step 6 and calculate the result for x.

x = (50 * tan(21 degrees)) + (50 * tan(39 degrees))

After evaluating this expression, you should get the distance between the tops of the two trees.

Note: Since this distance was asked to be rounded off to the nearest meter, be sure to round your final answer accordingly.