The drawing shows a person looking at a building on top of which an antenna is mounted. The horizontal distance between the person’s eyes and the building is 87.8 m. In part a the person is looking at the base of the antenna, and his line of sight makes an angle of 30.3o with the horizontal. In part b the person is looking at the top of the antenna, and his line of sight makes an angle of 33.8o with the horizontal. How tall is the antenna?

To find the height of the antenna, we can use trigonometry and the concept of similar triangles.

Let's start by labeling the given values and unknown values on the diagram.
- The horizontal distance between the person's eyes and the building is given as 87.8 m.
- In part a, the angle between the line of sight and the horizontal is given as 30.3 degrees.
- In part b, the angle between the line of sight and the horizontal is given as 33.8 degrees.
- The height of the antenna is the unknown value we're trying to find.

Now, let's break down the problem into two parts: finding the height of the building and finding the height of the antenna.

Part 1: Finding the height of the building
In part a, the person is looking at the base of the antenna, so the vertical distance between the person's eyes and the base of the antenna is the height of the building.
Using trigonometry, we can set up the following equation:
tan(30.3°) = height of the building / 87.8 m

Simplifying the equation:
height of the building = 87.8 m * tan(30.3°)

Part 2: Finding the height of the antenna
In part b, the person is looking at the top of the antenna, so the vertical distance between the person's eyes and the top of the antenna is the height of the building plus the height of the antenna.
Using trigonometry, we can set up the following equation:
tan(33.8°) = (height of the building + height of the antenna) / 87.8 m

Simplifying the equation:
height of the building + height of the antenna = 87.8 m * tan(33.8°)

Now, we can substitute the value we found in Part 1 for the height of the building into the equation in Part 2 and solve for the height of the antenna:
87.8 m * tan(33.8°) = (87.8 m * tan(30.3°)) + height of the antenna

Rearranging the equation to solve for the height of the antenna:
(height of the antenna) = (87.8 m * tan(33.8°)) - (87.8 m * tan(30.3°))

Calculating the value:
(height of the antenna) ≈ 87.8 m * (tan(33.8°) - tan(30.3°))

Now, plug the values into a calculator to find the final answer for the height of the antenna. Round the result to an appropriate number of decimal places based on the given information and the required precision.

To find the height of the antenna, we can use trigonometry and the given information.

Let's start by analyzing part a, where the person is looking at the base of the antenna.

In this case, we know the following:
- Horizontal distance between the person's eyes and the building = 87.8 m
- Angle of the line of sight with the horizontal = 30.3 degrees

We can define the following right-angled triangle:

|\
| \
| \
| \
| \
Height (h) | \ Base of
| \ Antenna
| \
__________________|_________\
87.8 m

Since we want to find the height (h) of the antenna, we'll focus on the opposite side of the triangle.

Using trigonometry, we can write:

tan(angle) = opposite / adjacent

tan(30.3 degrees) = h / 87.8 m

Rearranging the equation to solve for h:

h = tan(30.3 degrees) * 87.8 m

Calculating the value of h:

h = 0.5892 * 87.8 m

h ≈ 51.69 m

So, the height of the antenna in part a is approximately 51.69 m.

Now, let's move on to part b, where the person is looking at the top of the antenna.

In this case, we know the following:
- Horizontal distance between the person's eyes and the building = 87.8 m
- Angle of the line of sight with the horizontal = 33.8 degrees

We can define the following right-angled triangle:

Height (h) |\
| \
| \
| \
| \
| \
__________________|______\
87.8 m

This time, we are interested in the total height, which includes the height calculated in part a (51.69 m) and the additional height from part b.

Using trigonometry, we can write:

tan(angle) = opposite / adjacent

tan(33.8 degrees) = (h + 51.69 m) / 87.8 m

Rearranging the equation to solve for h:

h + 51.69 m = tan(33.8 degrees) * 87.8 m

Calculating the value of h:

h = (tan(33.8 degrees) * 87.8 m) - 51.69 m

h ≈ 39.82 m

So, the additional height of the antenna in part b is approximately 39.82 m.

To find the total height of the antenna, we add the height calculated in part a to the additional height calculated in part b:

Total height of the antenna = 51.69 m + 39.82 m

Total height of the antenna ≈ 91.51 m

Therefore, the height of the antenna is approximately 91.51 m.