From a point A, 30 meters from the base of a building B, the angle of elevationto the top of the building C is 56 degrees. The angle of elevation to the top of flagpole D on top of the building is 60 degrees. Find the length of flagpole CD
Did you make a sketch?
I followed your labels and let the base of the building be B
in triangle DAB, tan30 = DB/30
DB = 30tan30° = ...
in triangle CAB, tan 60 = CB/30
CB = 30tan60 = ...
CD = CB - DB = ....
To find the length of flagpole CD, we can use the trigonometric ratios. Let's break down the problem into smaller steps:
Step 1: Visualize the problem.
Draw a diagram to better understand the situation. Label the given information as follows:
- Point A is the location where we are standing.
- Point B is the base of the building.
- Point C is the top of the building.
- Point D is the top of the flagpole.
- Angle ABC represents the angle of elevation from point A to the top of the building.
- Angle ACD represents the angle of elevation from point A to the top of the flagpole.
- The distance AB is given as 30 meters.
Step 2: Identify the triangle and the trigonometric ratio.
In this case, we have a right-angled triangle ACD, with angle ACD being the angle between the perpendicular side CD and the hypotenuse AC. We need to find the length of the side CD.
Step 3: Determine the trigonometric ratio.
Since we are given the angle ACD, which is 60 degrees, we can use the sine ratio, which is opposite/hypotenuse. In this case, sin(60) = CD/AC.
Step 4: Find the length of AC.
To find the length of AC, we can use the trigonometric ratio in the triangle ABC. Since we know the angle ABC is 56 degrees and the length AB is given as 30 meters, we can use the tangent ratio, which is opposite/adjacent. In this case, tan(56) = BC/AB. Rearranging the equation, BC = tan(56) * AB.
Step 5: Substitute the values and solve the equations.
We can now substitute the values we have into the equations we derived:
- AC = BC + AB (using the Pythagorean theorem since we have a right-angled triangle ABC)
- sin(60) = CD/AC
- tan(56) = BC/AB
Solving these equations will give us the length of CD. Let's calculate it step by step:
- Step 1: BC = tan(56) * AB
BC = tan(56) * 30
BC ≈ 46.89 meters (rounded to two decimal places)
- Step 2: AC^2 = BC^2 + AB^2
AC^2 = (46.89)^2 + 30^2
AC^2 ≈ 2970.52 + 900
AC^2 ≈ 3870.52
AC ≈ √3870.52
AC ≈ 62.22 meters (rounded to two decimal places)
- Step 3: sin(60) = CD/AC
0.866 = CD/62.22
CD = 0.866 * 62.22
CD ≈ 53.90 meters (rounded to two decimal places)
Therefore, the length of the flagpole CD is approximately 53.90 meters.