A man standing 9 meters above the ground observes the angles of elevation and depression of the top and bottom,of the top of the monument in luneta 6 degrees and 50 minutes and 7 degrees and 30 minutes respectively.Find the height of the monument?
To solve this problem, we will use the concept of trigonometry and set up two equations based on the angles of elevation and depression.
Let's assume the height of the monument is 'h' meters.
From the given information, we can set up the following equation based on the angle of elevation:
tan(6° 50') = h / x, (equation 1)
where 'x' is the distance between the man and the monument at the bottom.
Similarly, we can set up another equation based on the angle of depression:
tan(7° 30') = (h - 9) / x, (equation 2)
Now, let’s solve these equations simultaneously to find the height of the monument:
Rearranging equation 1, we get:
x = h / tan(6° 50').
Substituting this value in equation 2, we have:
tan(7° 30') = (h - 9) / (h / tan(6° 50')).
Now, we can solve for 'h'.
First, convert the angles from degrees and minutes to decimal form:
6° 50' = 6 + (50 / 60) = 6.8333°,
7° 30' = 7 + (30 / 60) = 7.5°.
Substituting these values, the equation becomes:
tan(7.5°) = (h - 9) / (h / tan(6.8333°)).
Using a calculator or online trigonometric calculator, we can find:
tan(7.5°) ≈ 0.1317,
tan(6.8333°) ≈ 0.1197.
Now, we can substitute these approximate values in the equation:
0.1317 ≈ (h - 9) / (h / 0.1197).
To solve for 'h', cross multiply and simplify:
0.1317 * (h / 0.1197) = h - 9.
Divide both sides of the equation by 0.1197:
h * 1.0992 ≈ h - 9.
Subtract 'h' from both sides of the equation:
h * 0.0992 ≈ -9.
Finally, divide both sides of the equation by 0.0992 to solve for 'h':
h ≈ -9 / 0.0992.
Using a calculator, we find:
h ≈ -90.7258 meters.
Since height cannot be negative, we discard the negative value and conclude that the height of the monument is approximately 90.73 meters.
To find the height of the monument, we can use the tangent function, which relates the angle of elevation or depression to the height and distance.
Let's label the height of the monument as h and the distance from the man's position to the base of the monument as d.
Based on the given information, we know that the angle of elevation from the man's position to the top of the monument is 6 degrees and 50 minutes. We also know that the angle of depression from the man's position to the bottom of the monument is 7 degrees and 30 minutes.
To solve the problem, follow these steps:
Step 1: Convert the angles of elevation and depression from degrees and minutes to decimal form.
Convert 6 degrees and 50 minutes to decimal form:
6 degrees + 50 minutes / 60 = 6 + 50/60 = 6 + 5/6 = 6.0833 degrees
Convert 7 degrees and 30 minutes to decimal form:
7 degrees + 30 minutes / 60 = 7 + 30/60 = 7 + 1/2 = 7.5 degrees
Step 2: Use the tangent function to set up two equations to find h and d.
For the angle of elevation:
tan(angle of elevation) = h / d -> tan(6.0833 degrees) = h / d
For the angle of depression:
tan(angle of depression) = h / (d + 9) -> tan(7.5 degrees) = h / (d + 9)
Step 3: Solve the equations simultaneously to find the values of h and d.
tan(6.0833 degrees) = h / d
tan(7.5 degrees) = h / (d + 9)
Rearranging the equations, we get:
h = d * tan(6.0833 degrees)
h = (d + 9) * tan(7.5 degrees)
Set the two equations equal to each other:
d * tan(6.0833 degrees) = (d + 9) * tan(7.5 degrees)
Now, solve this equation for d.
Step 4: Calculate the height of the monument.
Plug the value of d back into one of the original equations to find h.
h = d * tan(6.0833 degrees)
So, by substituting the value of d into this equation, we can find the height of the monument.