Sure, let's go through the calculation step by step to find the height of the flagstaff on top of the tower.
### Step 1: Understand the Problem
You are given two angles of elevation from a point on the ground to the top and bottom of a flagstaff on a tower:
- The angle to the top of the flagstaff: 64°
- The angle to the bottom of the flagstaff: 62°
The distance from the observer to the base of the tower is not explicitly given, so we will assume a distance of 50 meters for the sake of example.
### Step 2: Convert Angles from Degrees to Radians
To perform trigonometric calculations, we need to convert the angles from degrees to radians.
- Angle to the top of the flagstaff in radians = 64° = 1.1170 radians (approximately)
- Angle to the bottom of the flagstaff in radians = 62° = 1.0821 radians (approximately)
### Step 3: Calculate the Height from the Base to the Bottom of the Flagstaff
Using trigonometry, specifically the tangent function (which is the ratio of the opposite side to the adjacent side), we can calculate the height from the base of the tower to the bottom of the flagstaff:
[tex]\[ \text{Height}_{\text{bottom}} = \text{distance} \times \tan(\text{angle}_{\text{bottom}}) \][/tex]
[tex]\[ \text{Height}_{\text{bottom}} = 50 \times \tan(1.0821) \][/tex]
[tex]\[ \text{Height}_{\text{bottom}} = 94.0363 \text{ meters} \][/tex]
### Step 4: Calculate the Height from the Base to the Top of the Flagstaff
Similarly, we calculate the height from the base of the tower to the top of the flagstaff:
[tex]\[ \text{Height}_{\text{top}} = \text{distance} \times \tan(\text{angle}_{\text{top}}) \][/tex]
[tex]\[ \text{Height}_{\text{top}} = 50 \times \tan(1.1170) \][/tex]
[tex]\[ \text{Height}_{\text{top}} = 102.5152 \text{ meters} \][/tex]
### Step 5: Calculate the Height of the Flagstaff
The height of the flagstaff is the difference between the height to the top of the flagstaff and the height to the bottom of the flagstaff:
[tex]\[ \text{Height}_{\text{flagstaff}} = \text{Height}_{\text{top}} - \text{Height}_{\text{bottom}} \][/tex]
[tex]\[ \text{Height}_{\text{flagstaff}} = 102.5152 - 94.0363 \][/tex]
[tex]\[ \text{Height}_{\text{flagstaff}} = 8.4789 \text{ meters} \][/tex]
### Step 6: Round the Height of the Flagstaff to One Decimal Place
[tex]\[ \text{Height}_{\text{flagstaff}} \approx 8.5 \text{ meters} \][/tex]
So, the height of the flagstaff, correct to one decimal place, is 8.5 meters.
