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 am really confused by this question like I know what its asking but I don't know how to draw it out on paper so i can work it out.

Draw a "side view"

Draw 3 points on the river, B , C, and D , B to the left of the points
draw a point A above B so that angle B is 90°
Join AC and AC , so that CD = 20
Mark angle ADC = 62.6° and angle ACB = 72.8°
label AB =h (h is the height) , BC = x

in triangle ABC:
tan 72.8° = h/x
h = xtan72.8°

in triangle ABD
tan62.6° = h/(x+20)
h = (x+20)tan62.6°

xtan72.8 = (x+20)tan62.6
xtan72.8 - xtan62.6) = 20tan62.6
x(tan72.8 - tan62.6) = 20tan62.6
x = 20tan62.6/(tan72.8 - tan62.6)

sub back into h = xtan72.8
h = 20(tan62.6)(tan72.8)/(tan72.8-tan62.6)
= ..... you do the button pushing

notice up to this point, I have not done any arithmetic or calculator work.
Also notice that the fact that the bridge was 1 km long did not enter the picture. It is important that you recognize what information is relevant and what is not.
BTW, I got 95.79 m

Alternate way:

in triangle ACD (not right-angled) , angle A = 10.2° , angle C = 107.2°
By the Sine Law:
AC/sin62.6 = 20/sin10.2
AC = 20sin62.5/sin10.2

in the right-angled triangle ABC
sin72.8 = h/AC
h = AC sin72.8 = (20sin62.6/sin10.2)(sin72.8) = 95.79

HOW ABOUT THAT ??

Thanks so much!

Well, don't worry too much about drawing it out on paper. I'm here to help you figure it out with words (and humor, of course!). Let's break it down step by step.

First, let's assume that the distance from the first point on the river to the bridge is x meters. So, from the first point, the height of the bridge above the river is h meters.

Now, using some trigonometry magic, we can say that tan(62.6°) = h / x. Similarly, tan(72.8°) = h / (x - 20).

We can simplify this further. Tan-rum-pum-pum! (Oops, sorry, I couldn't resist the temptation for a little drum roll.)

Multiplying both sides of the first equation by x, we get h = x * tan(62.6°). And multiplying both sides of the second equation by (x - 20), we get h = (x - 20) * tan(72.8°).

Now, we have two equations for h. Aren't we lucky?

So, x * tan(62.6°) = (x - 20) * tan(72.8°).

Now, let's solve this equation. But don't be scared! I won't be drawing any clowns on this one.

Expanding the equation will give you x * tan(62.6°) - 20 * tan(72.8°) = x * tan (72.8°).

Rearrange the equation to isolate x: x * (tan(62.6°) - tan(72.8°)) = 20 * tan(72.8°).

Finally, divide both sides by (tan(62.6°) - tan(72.8°)) to solve for x. And there you have it!

Once you have the value for x, you can substitute it back into either of the original equations to find h, the height of the bridge above the river.

Now, I could give you the exact numerical answer, but hey, let's embrace some suspense, shall we? I'll let you plug in the numbers and calculate it yourself. Trust me, it will be a bridge-tastic experience!

To solve this problem, we can break it down into smaller steps:

Step 1: Draw a Diagram
Start by drawing a rough sketch of the situation described in the question. Draw a horizontal line to represent the river, and above it, draw a diagonal line to represent the high-level bridge. Label the two points mentioned in the question: one point closer to the bridge and the other point further away.

Step 2: Label the Distances
Label the distance between the first point and the bridge as "x" (in meters) and the distance between the second point and the bridge as "x + 20" meters. Since the difference in distance between the two points is given as 20m, we can use this information.

Step 3: Identify the Angles
Based on the information provided, we have two angles of elevation: 62.6 degrees and 72.8 degrees. Label these angles in your diagram.

Step 4: Identify the Height of the Bridge
Let's assume that the height of the bridge above the river is "h" (in meters).

Step 5: Trigonometric Relationships
Applying trigonometry, we can use the tangent function to relate the angles of elevation to the height and distances. The tangent of an angle is equal to the opposite side divided by the adjacent side.

For the first point (angle of elevation = 62.6 degrees):
tan(62.6) = h / x

For the second point (angle of elevation = 72.8 degrees):
tan(72.8) = h / (x + 20)

Step 6: Solve the Equations
Now, you have two equations with two variables (h and x). Solve these equations simultaneously to find the value of h.

From the first equation:
tan(62.6) = h / x
x = h / tan(62.6) -> Equation 1

From the second equation:
tan(72.8) = h / (x + 20)
(x + 20) = h / tan(72.8) -> Equation 2

Now, we have a system of equations (Equation 1 and Equation 2). We can solve these equations using substitution or elimination method.

Step 7: Substitute or Eliminate
Substitute Equation 1 into Equation 2:
(h / tan(62.6) + 20) = h / tan(72.8)

Step 8: Solve for h
Now, solve the equation for h by isolating it.

tan(72.8) * (h / tan(62.6) + 20) = h
tan(72.8) * h / tan(62.6) + 20 * tan(72.8) = h
h / tan(62.6) * tan(72.8) + 20 * tan(72.8) = h
h * (tan(72.8) / tan(62.6) - 1) = -20 * tan(72.8)
h = -20 * tan(72.8) / (tan(72.8) / tan (62.6) - 1)

Step 9: Calculation
Using a scientific calculator, evaluate the expression above to find the value of h.

Step 10: Round the Final Answer
Once you have calculated the value of h from Step 9, round it to the nearest meter as per the question.

That's it! You have now determined the height of the bridge to the nearest meter.

To solve this problem, it is helpful to draw a diagram. Here are the steps to draw the diagram to help you visualize the problem:

1. Draw a straight horizontal line to represent the surface of the river.
2. Label one end of the line as the starting point and the other end as the bridge.
3. Draw a vertical line at the starting point to represent a point of observation on the river.
4. Label this point as "Observer 1".
5. Draw another vertical line 20m closer to the bridge from the previous line.
6. Label this point as "Observer 2".
7. Draw a line from the top of the bridge to "Observer 1".
8. Label the angle between the line representing the river and the line representing the bridge as 62.6 degrees.
9. Draw another line from the top of the bridge to "Observer 2".
10. Label the angle between the line representing the river and the line representing the bridge as 72.8 degrees.

Now that you have visualized the problem, let's proceed to solve it.

To find the height of the bridge, we can use the concept of trigonometry. Specifically, we'll use the tangent function. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side of a right triangle.

In this case, the vertical distance from either observer to the top of the bridge is the opposite side, and the horizontal distance between the observers is the adjacent side.

Let's denote the height of the bridge as "h" and the shared horizontal distance between the observers as "d". Based on the diagram, we can see that the equation for Observer 1 is:

tan(62.6 degrees) = h / d

Similarly, for Observer 2, the equation becomes:

tan(72.8 degrees) = h / (d - 20)

Now, we have two equations with two variables. By solving these equations simultaneously, we can find the values of "h" and "d". Once we know "h", we can determine the height of the bridge.

Please note that to calculate the exact values, you would need to use a scientific calculator or software capable of handling trigonometric functions.