to calculate the height of the tower david measured the angle of elevation of the top of the tower from point A to B 42degrees. He moved then 30meter closer to the tower and from the point B the angle of elevation to the top of the tower to be 50degree. Determine the height of the tower
Did you make a sketch?
I labeled the tower as PQ, with Q and the extended line of AB
In triangle PAB
angle A = 42°, angle ABP = 130°
then angle APB = 8°
by sine law:
BP/sin42 = 30/sin8
BP = 30sin42/sin8
In the right-angled triangle BPQ
sin50 = PQ/BP
PQ = BPsin50
= (30sin42/sin8)(sin50)
carry on with the button-pushing
To find the height of the tower, we can use the trigonometric formula for the tangent of an angle.
Let's denote the height of the tower as "h" and the distance between point A and the tower as "x".
Step 1: Determine the distance between point B and the tower.
Since David moved 30 meters closer to the tower from point A, the distance between point B and the tower would be x - 30.
Step 2: Use the tangent function to set up an equation.
From point A, the tangent of the angle can be written as:
tan(42°) = h / x
From point B, the tangent of the angle can be written as:
tan(50°) = h / (x - 30)
Step 3: Solve the equation for "h".
Using the value of the tangent function, we can set up the equation as follows:
tan(42°) = h / x
tan(50°) = h / (x - 30)
Step 4: Solve for "h".
Solving the equations simultaneously will give us the value for "h". We can rearrange the first equation to express x in terms of h:
x = h / tan(42°)
Using this expression, substitute it into the second equation:
tan(50°) = h / (h / tan(42°) - 30)
Simplify the equation:
tan(50°) = h / (h / tan(42°) - 30)
tan(50°) = tan(42°) * h / (h - 30 * tan(42°))
Cross-multiplying leads to:
tan(50°) * (h - 30 * tan(42°)) = tan(42°) * h
Expand:
h * tan(50°) - 30 * tan(42°) * tan(50°) = h * tan(42°)
Rearrange terms:
h * (tan(50°) - tan(42°))) = 30 * tan(42°) * tan(50°)
Finally, solve for "h":
h = (30 * tan(42°) * tan(50°)) / (tan(50°) - tan(42°))
Using a calculator, calculate the value of "h".
To determine the height of the tower, we can use trigonometry. Let's break down the problem step by step.
First, let's label the relevant parts of the problem:
- Point A represents the initial position where David measured the angle of elevation.
- Point B represents the new position after David moved 30 meters closer to the tower.
- The angle of elevation from point A to the top of the tower is 42 degrees.
- The angle of elevation from point B to the top of the tower is 50 degrees.
Let's assume the height of the tower is represented by "h."
Now, let's consider the right triangle formed by the tower, point A, and the ground. In this triangle:
- The opposite side is the height of the tower, "h."
- The adjacent side is the distance from point A to the tower (which we don't know yet).
- The angle opposite the height of the tower is 42 degrees.
Using the tangent function, we can set up the following equation:
tan(42 degrees) = h / (distance from A to the tower)
Next, let's consider the right triangle formed by the tower, point B, and the ground. In this triangle:
- The opposite side is still the height of the tower, "h."
- The adjacent side is the distance from point B to the tower plus the 30 meters David moved closer.
- The angle opposite the height of the tower is 50 degrees.
Again, using the tangent function, we can set up the following equation:
tan(50 degrees) = h / (distance from B to the tower + 30 meters)
Now, we have two equations with two unknowns: the height of the tower, "h," and the distance from A to the tower. We can solve these equations simultaneously to find the height of the tower.
Once the values for the distances from A to the tower and B to the tower are found, the height of the tower can be determined by substituting one of the distances into either of the equations.