from point on horizontal ground a surveyor measures the angle of elevation of the top of a flagpole as18 degrees 40feets he moves 50meters nearer to the flagpole and measures the angle of elevation as degrees26
22feets.determine the height the height of the flagpole
I didn't understand
To find the height of the flagpole, we can use trigonometry and set up a right triangle.
Let's label the following information:
- The angle of elevation from the initial position is 18 degrees.
- The angle of elevation from the second position is 26 degrees.
- The distance between the initial position and the flagpole is x meters.
- The height of the flagpole is h feet.
Step 1: Calculating the height from the initial position
In the right triangle formed by the initial position, the flagpole, and the point on the ground, we can use the tangent function:
tan(18 degrees) = h / x
Step 2: Calculating the height from the second position
In the right triangle formed by the second position, the flagpole, and the same point on the ground, we can again use the tangent function:
tan(26 degrees) = h / (x - 50 meters)
Step 3: Solving the equations
Now, we have two equations with two unknowns (h and x). We can solve this system of equations.
First, rewrite the equations:
1) tan(18 degrees) = h / x
2) tan(26 degrees) = h / (x - 50 meters)
Next, substitute the values of the tangent functions:
1) tan(18 degrees) = h / x
2) tan(26 degrees) = h / (x - 50)
Now, we can solve for x by cross-multiplying:
1) x = h / tan(18 degrees)
2) (x - 50) = h / tan(26 degrees)
Since both equations are equal to x, we can set them equal to each other:
h / tan(18 degrees) = h / tan(26 degrees) + 50
Finally, we can solve for h:
h / tan(18 degrees) - h / tan(26 degrees) = 50
Multiply through by the common denominator (tan(18 degrees) * tan(26 degrees)):
(tan(26 degrees) * h - tan(18 degrees) * h) / (tan(18 degrees) * tan(26 degrees)) = 50
Simplify and solve for h:
h * (tan(26 degrees) - tan(18 degrees)) = 50 * (tan(18 degrees) * tan(26 degrees))
h = (50 * (tan(18 degrees) * tan(26 degrees))) / (tan(26 degrees) - tan(18 degrees))
Using a scientific calculator or trigonometric tables, we can find the values of tan(18 degrees) and tan(26 degrees). Plugging in these values will provide the final result for h, the height of the flagpole.
To determine the height of the flagpole, we can use trigonometry and create two right triangles.
Let's label the important points:
A -> Initial position of the surveyor
B -> Final position of the surveyor
C -> Top of the flagpole
We can create two right triangles: Triangle ABC and Triangle ABD. The side opposite the angle of elevation in both triangles will be the height of the flagpole.
In Triangle ABC:
Angle BAC is the angle of elevation of the top of the flagpole, which is 18 degrees.
Side AC is the height of the flagpole, which we want to find.
In Triangle ABD:
Angle BAD is the angle of elevation of the top of the flagpole from the new position, which is 26 degrees.
Side AC is the height of the flagpole, which we want to find.
BD is the distance the surveyor moved nearer to the flagpole, which is 50 meters (given).
Now, let's use trigonometry to write equations for each triangle:
In Triangle ABC:
tan(18 degrees) = AC / AB
In Triangle ABD:
tan(26 degrees) = AC / (AB - 50)
We can rearrange both equations to solve for AC, which is the height of the flagpole:
AC = tan(18 degrees) * AB
AC = tan(26 degrees) * (AB - 50)
Now, we have two equations with two unknowns (AC and AB). We can solve this system of equations to find the height of the flagpole.
Divide the second equation by the first equation to eliminate AB:
(tan(26 degrees) * (AB - 50)) / (tan(18 degrees) * AB) = AC / AC
tan(26 degrees) / tan(18 degrees) = AC / AC
tan(26 degrees) / tan(18 degrees) = 1
tan(26 degrees) / tan(18 degrees) = 1
Now, we can solve this equation to find the value of tan(26 degrees) / tan(18 degrees). Use a scientific calculator to find the value, which is approximately 1.5095.
So, the height of the flagpole (AC) is equal to 1.5095 times the distance from the initial position of the surveyor to the flagpole (AB).
Now, let's substitute this value back into one of the equations:
AC = tan(18 degrees) * AB
AC = 1.5095 * AB
We still have one unknown (AB). To find it, let's substitute the given information into the equation:
40 feet = 1.5095 * AB
Divide both sides by 1.5095:
AB ≈ 40 feet / 1.5095
AB ≈ 26.51 feet
So, the distance from the initial position of the surveyor to the flagpole (AB) is approximately 26.51 feet.
Now, substitute this value back into the equation to find the height of the flagpole:
AC = 1.5095 * AB
AC = 1.5095 * 26.51 feet
AC ≈ 40 feet
Therefore, the height of the flagpole is approximately 40 feet.
draw a diagram
review your basic trig functions
now it is clear that you can find the height h using
h cot 18°40' - h cot 26°22' = 50