From a point x = 85 feet in front of a public library, the angles of elevation to the base of the flagpole and the top of the flagpole are θ = 33.5° and 39° 45', respectively. The flagpole is mounted on the front of the library's roof. Find the height of the flagpole.
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STEP 1: Let F be the height of the flagpole and L be the height of the public library. Convert the angle 39° 45' into decimal form.
Use this value and an appropriate trigonometric function to find F + L, the total height of the library and flagpole.
STEP 2: Find the height of the library, L.
STEP 3: Subtract your result from Step 2 from your latter result in Step 1 to determine the height of the flagpole, F.
Did you use
L = 85tan 33.5 and L+F = 85 tan 39.5 to find F ?
To solve this problem, we can use the concept of trigonometry and right triangles. We will use the tangent function to find the height of the flagpole.
Let's start by labeling the given information:
x = distance from the point to the library = 85 feet
θ1 = angle of elevation to the base of the flagpole = 33.5°
θ2 = angle of elevation to the top of the flagpole = 39° 45'
Now, let's find the height of the flagpole step by step:
Step 1: Convert the angle θ2 from degrees and minutes to decimal form:
θ2 = 39° 45' = 39 + (45/60) = 39.75°
Step 2: Use the tangent function to find the height of the flagpole using θ1 and x:
tan(θ1) = height of the flagpole / x
height of the flagpole = x * tan(θ1)
Plugging in the values:
height of the flagpole = 85 * tan(33.5°)
Step 3: Calculate the height of the flagpole using a calculator:
height of the flagpole ≈ 85 * 0.684 = 58.14 feet
So, the height of the flagpole is approximately 58.14 feet.
To find the height of the flagpole, we can use trigonometry and the given angles of elevation. Let's break down the steps to solve the problem:
Step 1: Draw a diagram:
It's essential to visualize the scenario described in the problem. Draw a diagram with a horizontal line to represent the ground, a vertical line for the flagpole, and a slanted line connecting the point x (85 feet from the library) to the top of the flagpole.
Step 2: Determine the right triangle:
Identify the right triangle formed by the horizontal line (ground), the slanted line (85 feet), and the vertical line (flagpole's height). This right triangle exhibits the angles of elevation given in the problem.
Step 3: Calculate the height of the flagpole:
Using trigonometric ratios, we'll focus on the angle of elevation of 33.5°.
Since tan(θ) = opposite/adjacent, we have:
tan(33.5°) = height of flagpole / 85 feet
Rearranging the equation, we get:
height of flagpole = 85 feet * tan(33.5°)
Calculating this expression gives us the height of the flagpole.
Step 4: Convert the angle of elevation to degrees:
The angle of elevation for the top of the flagpole is given as 39° 45', but we need to convert it to decimal degrees.
To convert from degrees, minutes, and seconds to decimal degrees, we can use the following formula:
decimal degrees = degrees + (minutes/60) + (seconds/3600)
For the given angle of 39° 45', the decimal degrees would be:
39 + (45/60) = 39.75°
Step 5: Repeat step 3 for the top of the flagpole:
Using the same trigonometric ratio, but this time with the angle of 39.75°, we can calculate the height of the flagpole from the top to the bottom:
height of flagpole = 85 feet * tan(39.75°)
This expression will give us the height of the flagpole from the top down to the roof.
Step 6: Calculate the total height of the flagpole:
To get the complete height of the flagpole, we need to add the heights calculated in steps 3 and 5:
Total height of flagpole = (height of flagpole using θ=33.5°) + (height of flagpole using θ=39.75°)
By following these steps, you can find the height of the flagpole based on the given information and trigonometric calculations.