Cand D are two observation posts on the same horizontal ground at the foot A of a vertical tower AB.the tower is 18m due north of D and 24m east of C.the angle of elevation of B from D is 35¡ã.calculate to three s.f the height AB,distance CD,angle of elevation from C,bearing...
As with all geometry problems, draw a diagram (plan).
You will find that in right triangle ABD, we know
AD=18 m
∠ADB=35°
Solve this triangle to get
mAB=AD tan(ADB)
=18tan(35°)
For right triangle CAB,
mAB (known from above)
mCA = 24m
=>
∠ACB=tan-1(mAB/mCA)
=angle of elevation of B relative to C.
Distance CD can be found by Pythagoras theorem.
Why did the observation posts go to the same horizontal ground?
Because they wanted to have a level playing field!
Now, let's solve your problem.
We have a right-angled triangle with sides AB, CB, and AD.
We know that AD = 18m and CB = 24m.
To find AB, we can use the Pythagorean theorem:
AB² = AD² + DB²
Since DB is the hypotenuse of a right-angled triangle with an elevation angle of 35°,
we can use trigonometry to find its length:
DB = AD / tan(35°)
Now we can calculate AB:
AB² = (18m)² + (AD / tan(35°))²
AB ≈ √(18m)² + ((18m) / tan(35°))²
AB ≈ √(324m²) + (116.108m)²
AB ≈ √105264m² + 13478.887m²
AB ≈ √118742.89m²
AB ≈ 344.75m
So, the height AB is approximately 344.75m.
To find the distance CD, we can use the Pythagorean theorem again:
CD² = CB² + DB²
CD ≈ √(24m)² + ((18m) / tan(35°))²
CD ≈ √576m² + 13478.887m²
CD ≈ √14016m² + 13478.887m²
CD ≈ √27494.888m²
CD ≈ 165.71m
So, the distance CD is approximately 165.71m.
Now, let's find the angle of elevation from C:
tan(angle of elevation from C) = AB / CB
tan(angle of elevation from C) ≈ 344.75m / 24m
angle of elevation from C ≈ arctan(344.75m / 24m)
angle of elevation from C ≈ 86.81°
Lastly, let's determine the bearing:
The bearing is the angle measured clockwise from the true north.
In this case, the tower is 18m due north of D, so the bearing is 0°.
To solve this problem, we'll use trigonometric ratios and basic geometry concepts.
Step 1: Calculate the height AB:
Let's consider triangle ADB. The angle of elevation of B from D is 35°. Since the angle of elevation is the angle between the horizontal ground and the line of sight from D to B, we can consider the triangle ADB to be a right triangle.
Using the tangent ratio, we have:
tan(35°) = AB / AD
AB = AD * tan(35°)
AB = 18 * tan(35°)
AB ≈ 11.92 m
So, the height AB is approximately 11.92 meters.
Step 2: Calculate the distance CD:
We can consider the triangle CDB as a right triangle.
Let x be the distance CD. Then, the length of BC will be 24 - x (since the tower is 24m east of C, and we're subtracting the distance CD to get the remaining distance BC).
Using the Pythagorean theorem, we have:
(x^2) + (BC^2) = (BD^2)
x^2 + (24 - x)^2 = 18^2
x^2 + 576 - 48x + x^2 = 324
2x^2 - 48x + 252 = 0
Solving this quadratic equation, we find two possible solutions:
x ≈ 4.76 or x ≈ 22.24
Since x represents a distance and cannot be negative, the distance CD is approximately 4.76 meters.
Step 3: Calculate the angle of elevation from C:
Let's consider triangle ABC. We want to find the angle of elevation of B from C.
Using the tangent ratio, we have:
tan(theta) = AB / BC
tan(theta) = 11.92 / (24 - 4.76)
tan(theta) = 11.92 / 19.24
theta ≈ tan^(-1)(11.92 / 19.24)
theta ≈ 34.27°
So, the angle of elevation from C is approximately 34.27°.
Step 4: Calculate the bearing:
The bearing is the direction measured in degrees clockwise from north.
To find the bearing from C to B, we can consider triangle CDB. The bearing will be the angle (α) measured between the line north and the line CD.
Using the cosine ratio, we have:
cos(α) = CD / BD
cos(α) = 4.76 / 18
α ≈ cos^(-1)(4.76 / 18)
α ≈ 74.14°
Since the bearing is measured clockwise from north, the bearing from C to B is approximately 360° - α.
Bearing ≈ 360° - 74.14°
Bearing ≈ 285.86°
So, the bearing is approximately 285.86°.
To summarize:
- The height AB is approximately 11.92 meters.
- The distance CD is approximately 4.76 meters.
- The angle of elevation from C is approximately 34.27°.
- The bearing from C to B is approximately 285.86°.
To solve this problem, we can use trigonometry and geometry concepts. Let's break it down step by step:
Step 1: Calculate the height AB:
We can use the tangent function to find the height of the tower. Let's define angle α as the angle of elevation from B to the top of the tower (angle BCB'). The tangent function is given by:
tan(α) = AB / CD
Rearranging the equation, we have:
AB = CD * tan(α)
Given that the angle of elevation from B to D is 35°, we can substitute the values:
AB = CD * tan(35°)
Step 2: Calculate the distance CD:
We can use the Pythagorean theorem to find the distance between the observation posts C and D. Let's define the horizontal distance from C to D as x and the vertical distance from C to A as y.
From the given information, we know that the distance AB = x + 24m and BD = y + 18m.
Using Pythagorean theorem, we have:
( x + 24 )^2 + ( y + 18 )^2 = CD^2
Step 3: Calculate the angle of elevation from C:
We can use the tangent function again to find the angle of elevation from C to the top of the tower (angle ACA'). The tangent function is given by:
tan(β) = AB / x
Rearranging the equation, we have:
β = atan(AB / x)
Step 4: Calculate the bearing:
The bearing is the angle formed clockwise from the north direction to a line. To calculate the bearing from C to B, we need to find the angle formed between the north direction and the line CB.
The bearing can be calculated using the inverse tangent function:
Bearing = atan(24 / y)
Now, let's calculate the values:
Step 1: Calculate the height AB:
AB = CD * tan(35°)
Step 2: Calculate the distance CD:
Solve the equation: ( x + 24 )^2 + ( y + 18 )^2 = CD^2
Step 3: Calculate the angle of elevation from C:
β = atan(AB / x)
Step 4: Calculate the bearing:
Bearing = atan(24 / y)
Now, plug in the values and calculate AB, CD, β, and the bearing using a calculator or software that supports trigonometric functions.