From a point 120 feet from the base of a plant building the angle of elevation to the roof line of the building is 38 degrees. The angle of elevation to the top of an antenna is 43 degrees. The antenna is attached to the nearest edge of the roof and the ground is flat. The height of the antenna is ________ feet. Round to two decimal places. I figured out that the height of the building, using the angle of 38 degrees, is 152.28 feet. I don't know how to figure the problem to solve for the height of the angennae.
To find the height of the antenna, we can use trigonometry and the information given in the problem. Let's break down the steps:
1. Draw a diagram: Start by drawing a diagram to visualize the situation. Draw a right-angled triangle with the following information:
- The base of the triangle represents the distance from the point to the building (120 feet).
- The angle of elevation to the roof line of the building is 38 degrees.
- The angle of elevation to the top of the antenna is 43 degrees.
2. Identify the relevant sides of the triangle: In this case, we are interested in finding the height of the antenna, so let's label it "h" (unknown). The side opposite the angle of elevation to the top of the antenna is also "h."
3. Determine the relationships between the sides and angles: We can use the trigonometric function tangent (tan) for this problem. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. For our triangle:
- tan(38 degrees) = height of the building (152.28 feet) / 120 feet
- tan(43 degrees) = height of the antenna / 120 feet
4. Solve for the unknown: Rearrange the second equation to solve for the height of the antenna (h):
- height of the antenna = tan(43 degrees) * 120 feet
5. Calculate the height: Use a calculator to find the value of tan(43 degrees) and then plug it into the formula:
- height of the antenna = (tan(43 degrees)) * 120 feet
Round the answer to two decimal places to get the final height of the antenna.