The high level bridge, a railway bridge that crosses the Oldman River is over 1km long. From one point on the river, the angle of elevation of the top of the bridge is 62.6 degrees. From a point 20m closer to the bridge, the angle of elevation of the top of the bridge is 72.8 degrees. How high is the brige above the river, to the nearest meter? i don't understand how to do this question.. please help!!:(
let h be height. Let d be the distance from the first observation to the bridge.
h/d=tan62.6
and
h/(d-20)=tan72.8
in the first equation
d=h ctn62.6
second equation
d=h ctn72.8 +20tan72.8
set these two equations equal, and solve for h.
or....
Consider the triangle formed with 20 m as the base and the two lines forming the angles of elevation to the top of the bridge.
62.6 will be the interior angle and 72.8 would be an exterior angle at the 20 m base.
That would make the interior angle at the base 107.2 and the small angle at the top where the lines meet at the bridge equal to 10.2 degrees.
by the Sine Law we can find the length of the line formed by the 72.8 degree line of observation.
this then becomes the hypotenuse of a right angled triangle where h is the height of the bridge, the side found above is the hypotenuse and the base angle is 72.8.
A simple sine ratio will find the height.
To solve this problem, we can break it down into two right triangles and use trigonometry. Let's label the points as follows:
Point A: The original point on the river where the angle of elevation is 62.6 degrees.
Point B: The point 20 m closer to the bridge where the angle of elevation is 72.8 degrees.
Point C: The top of the bridge.
Let's assume the height of the bridge is h meters.
Step 1: Calculate the distance from Point A to Point C:
tan(62.6 degrees) = h / x, where x is the distance from Point A to Point C.
x = h / tan(62.6 degrees) ---(1)
Step 2: Calculate the distance from Point B to Point C:
tan(72.8 degrees) = h / (x - 20), where x - 20 is the distance from Point B to Point C (20 m closer to the bridge).
x - 20 = h / tan(72.8 degrees) ---(2)
Step 3: Solve the equations (1) and (2) simultaneously to find the height of the bridge (h).
Let's solve the equations:
From equation (1):
x = h / tan(62.6 degrees) ---(3)
From equation (2):
x - 20 = h / tan(72.8 degrees) ---(4)
Substitute equation (3) into equation (4):
h / tan(62.6 degrees) - 20 = h / tan(72.8 degrees)
Multiply through by tan(62.6 degrees) * tan(72.8 degrees):
( tan(72.8 degrees) * h ) - ( 20 * tan(62.6 degrees) * h ) = 0
Factor out h:
h * ( tan(72.8 degrees) - 20 * tan(62.6 degrees) ) = 0
Divide both sides by ( tan(72.8 degrees) - 20 * tan(62.6 degrees) ):
h = 0 / ( tan(72.8 degrees) - 20 * tan(62.6 degrees) )
Simplify:
h = 0
It appears there was an error during calculations or data input. Please double-check the values and calculations.
To solve this question, we can use the concept of trigonometry, specifically the tangent function. Let's break down the problem step by step.
Step 1: Understanding the problem
We have two different locations along the river, from which we can measure the angle of elevation to the top of the bridge. By comparing these angles, we can determine the height of the bridge above the river.
Step 2: Visualize the problem
Draw a diagram to help visualize the situation. Label the top of the bridge as point A, the first observation point as B, the second observation point as C, and the river as O. Also, label the distance from B to A as x.
B C
|¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
|¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
|
|
|----¯O¯¯¯¯¯¯¯¯¯¯¯A
<-x->
Step 3: Identify known values
From the problem, we are given:
- Angle BOA (62.6 degrees)
- Angle COA (72.8 degrees)
- Distance from B to A (x = 20m)
Step 4: Identify what needs to be found
We need to find the height of the bridge above the river (OA).
Step 5: Apply trigonometry
Using the tangent function, we know that:
tan(angle) = opposite/adjacent
For triangle BOA:
tan(62.6) = OA/x
For triangle COA:
tan(72.8) = OA/(x-20)
Step 6: Solve the system of equations
We have two equations with two unknowns (OA and x). We can rearrange the equations to isolate OA and set them equal to each other:
OA/x = tan(62.6) (Equation 1)
OA/(x-20) = tan(72.8) (Equation 2)
Cross-multiplying Equation 1:
OA = x * tan(62.6)
Cross-multiplying Equation 2:
OA = (x-20) * tan(72.8)
Setting the two expressions for OA equal to each other:
x * tan(62.6) = (x-20) * tan(72.8)
Simplify and solve for x:
x * tan(62.6) = x * tan(72.8) - 20 * tan(72.8)
x * (tan(62.6) - tan(72.8)) = -20 * tan(72.8)
x = -20 * tan(72.8) / (tan(62.6) - tan(72.8))
Step 7: Calculate OA
Substitute the value of x back into either Equation 1 or Equation 2 to find the height of the bridge (OA).
OA = x * tan(62.6)
Step 8: Round the answer
Round the value of OA to the nearest meter, as the question requests.
And that's how you solve the problem! Remember to use a calculator to find the tangent of angles.