From the top of a building 85ft high, the angle of elevation of the top of a vertical pole is 11 degree 6'. At the bottom of the building the angle of elevation of the top of the pole is 26 degree 7'. Find the height of the pole and the distance of the pole from the following.
Draw a diagram. If the distance is x, then
(h-85)/x = tan11°6'
h/x = tan26°7'
So, (h-85)cot 11°6' = h*cot 26°7'
Use that to find h, and then you can get x.
To find the height of the pole and the distance from the pole, we can use trigonometric ratios and the concept of similar triangles.
Let's solve it step-by-step:
Step 1: Convert the angles from degrees and minutes to decimal form.
Angle of elevation at the top of the building = 11° 6'
Angle of elevation at the bottom of the building = 26° 7'
To convert degrees and minutes to decimal form, we can use the formula:
Decimal Angle = Degrees + (Minutes / 60)
Angle of elevation at the top of the building (decimal form):
Angle_top = 11 + (6 / 60) = 11.1°
Angle of elevation at the bottom of the building (decimal form):
Angle_bottom = 26 + (7 / 60) = 26.117° (rounded to 3 decimal places)
Step 2: Draw a diagram representing the given information.
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Let's assume:
- The height of the pole is h
- The distance from the top of the building to the base of the pole is x
We need to find the value of h and x.
Step 3: Using trigonometric ratios, we can set up two equations to solve for h and x.
In triangle AOB (where O is the top of the building and A is the bottom of the building):
tan(Angle_top) = h / x (Equation 1)
In triangle AOC (where C is the top of the pole):
tan(Angle_bottom) = (h + 85) / x (Equation 2)
Substitute the values of Angle_top and Angle_bottom into Equations 1 and 2:
tan(11.1°) = h / x
tan(26.117°) = (h + 85) / x
Step 4: Solve the equations simultaneously to find the values of h and x.
Rearrange Equation 1 to express h in terms of x:
h = x * tan(11.1°)
Substitute this value of h into Equation 2:
tan(26.117°) = (x * tan(11.1°) + 85) / x
Simplify the equation:
tan(26.117°) = tan(11.1°) + 85 / x
Multiply both sides by x:
x * tan(26.117°) = tan(11.1°) + 85
Rearrange the equation to solve for x:
x = (tan(11.1°) + 85) / tan(26.117°)
Substitute this value of x back into Equation 1 to find h:
h = x * tan(11.1°)
Calculate the values of x and h using a calculator:
x ≈ 225.94 ft (rounded to 2 decimal places)
h ≈ 45.7 ft (rounded to 1 decimal place)
Step 5: Interpret the results.
The height of the pole is approximately 45.7 ft.
The distance from the top of the building to the base of the pole is approximately 225.94 ft.
To solve this problem, we will apply trigonometric ratios and properties related to angles of elevation.
Step 1: Find the height of the pole
- Let's assume the height of the pole is 'h' feet.
- From the top of the building, the angle of elevation of the top of the pole is 11 degrees 6 minutes. This means that we have a right triangle where the opposite side is 'h' and the adjacent side is 85 ft.
- To find the height 'h', we need to use the tangent ratio: tan(angle) = opposite/adjacent.
- Plugging in the values, we have: tan(11° 6') = h/85.
- Compute the tangent of 11° 6' using a scientific calculator to get the value of tan(11° 6'). Multiply this value by 85 to find 'h'.
Step 2: Find the distance of the pole from the building at its bottom
- Let's assume the distance of the pole from the building is 'd' feet.
- At the bottom of the building, the angle of elevation of the top of the pole is 26 degrees 7 minutes. This gives us another right triangle where the opposite side is 'h' and the adjacent side is 'd'.
- To find the distance 'd', we will again use the tangent ratio: tan(angle) = opposite/adjacent.
- Plugging in the values, we have: tan(26° 7') = h/d.
- Compute the tangent of 26° 7' using a scientific calculator to get the value of tan(26° 7'). Multiply this value by 'h' to find 'd'.
By following these steps, you will be able to find the height of the pole and the distance of the pole from the building at its bottom.