trig help!??!?!?

Question

A bridges that crosses a river is 1km long. From 1 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 bridge above the water, to the nearest meter?

how would u solve this plz give thorough description

Answer 1

The height of the "bridges" is found as follows:

You make two right triangles, each with points at the bridge and directly below the bridge. The third point of each triangle are the two locations on the river that you're measuring from.

The first triangle has a base of x, the distance to the point directly underneath the bridge, a height of y, and an angle of 62.6 degrees.

The second triangle has a base of (x - 20), since you're 20 meters closer, a height of y, and an angle of 72.8 degrees.

In a crude drawing:

bridge
|
y
|
O <--x--> you

Where the angle is from point O, to you, to the bridge.

We know that the tangent of the angle equals y/x, so we solve each equation for x, in the hopes of canceling them out later:

tan(62.6 deg) = y / x
1.929 = y / x
x = y / 1.929

tan(72.8 deg) = y / (x - 20)
3.230 = y / (x - 20)
x - 20 = y / 3.230
x = y / 3.230 + 20

Now, we have two things that are both equal to x, so they must therefore be equal to each other. Then it's a simple matter of solving for y:

y / 1.929 = y / 3.230 + 20
0.5183*y = 0.3096*y + 20
0.2088*y = 20
y = 95.79

The height of the bridge over the water is therefore 95.79 meters.

Answer 2

Imagine two right triangles, one with a 62.6° angle and the other with a 72.8° angle which perfectly share the side that is opposite from these respective angles. You should see that the 72.8° triangle is inside the other and have sides adjacent to the angles which are connected. Also, their hypotenuses meet up at the same point.

Now, there are three lengths within this system which we want to pay attention to. One of them is the distance between the two points which have been labelled with angles. The length of the distance between these two points is 20m, as given by the problem.

Next, there is the distance between the point labelled 72.8° and the bottom point of its opposite triangle side. We will call this length x.

Finally, there is the side that is shared perfectly by both triangles. This is going to be the length that you are looking for. We can call this y.

All the rest is algebra and trigonometry.

Analyzing the picture, we can find two equations:
tan(62.6) = y/(20+x) which can be changed to y = (20+x)*tan(62.6)

and

tan(72.8) = y/x which can be changed to y = x*tan(72.8)

Putting these equations together gives us (20+x)*tan(62.6) = x*tan(72.8)

Solve for x:
x = 20*tan(62.6) / (tan[72.8] - tan[62.6]) and since I assume you are using a graphing calculator for this problem, I recommend that you store this number as something.

Now, plug this answer into one of the previous equations (either y = x*tan(72.8) or y = (20+x)*tan(62.6)) and there will be your answer for y.

Top bridge height is approximately 95.79 m

Answer 3

Make sketch as follows:-
AB is vertical line for bridge , B at river level.
C to left of B such that angle ACB = 72.8°
D to left of C such that angle ADB = 62.6°
DC = 20
CB = x
AB = h
tan 62.6° = h / (20 + x)
1.93 = h / (20 + x)
tan 72.8° = h / x
3.23 = h/x

38.6 + 1.93 x = h
3.23x = h

38.6 + 1.93 x = 3.23x
38.6 = 1.3 x
x = 29.7
tan 72.8° = h / 29.7
h = 29.7 x tan 72.8°
h = 95.9
Bridge is 95.9 m above water.

Answer 4

where is point 1 relative to the beginning of the bridge and where is point 2 relative to point 1