Capilano Suspension Bridge Problem

$$h=\dfrac{\sin{(41\ensuremath{^\circ} )}*\sin{(48\ensuremath{^\circ})}}{\sin{(41\ensuremath{^\circ}+48\ensuremath{^\circ})}}*140\;m = 68.27\;m$$

Hi Jedithious, it is great to see you again.

I can answer in full if you want but I am in a bit of a hurry right now so I will start with a condensed version.

You have an upside down triangle.  The top is the bridge, it is 140m long.  The apex at the bottom touches the water.

The perpendicular from the apex to the bridge is the height of the bridge.  Let that be h

The 2 angles at the top are 41 and 48 degrees.  I am assuming that the height of the bridge is the same on both sides.  this is important fo my solution.

So the perpendicular height cuts the bridge into 2 peices.  Let one peice be x metres and the other be 140-x metres.

Now you have tan 48=h/(140-x)  and tan 41=h/x

Make x the subject - put the 2 together and solve.

I haven't actually done this yet.  I don't know how difficult it is.  Have a play and get back to me.  I want to help you more if it is of any benefit to you.  This is a really good question.

♬ ♬ MELODY ♬ ♬

heureka's answer is a good one....how might we arrive at that????

We note that:

tan(41) = h/x     and that   tan(48) = h/(140-x)   where "x" is the horizontal distance from the side of the bridge where the 41 degree angle of depression is observed to the point where the perpendicular line representing the height intersects the bridge and  "h" is the height.

Using the second  equation, we can write h = (140-x)tan(48) and substituting this into the first equation for h, we have

tan(41) = (140-x)tan(48)/x    multiply both sides by x

xtan(41) = (140-x)tan(48)     simplifying the right, we have

xtan(41) = 140tan(48) - xian(48)   add xtan(48) to both sides

xtan(41) + xtan(48) = 140tan(48)   simplify the left side

x [tan(41) + tan(48)] = 140tan(48)   divide both sides by [tan(41) + tan(48)]

x = 140tan(48) / [tan(41) + tan(48)] 

And using the frist equation, we can write

h = xtan(41)

And substituting in for "x"  we have

h = 140tan(48) / [tan(41) + tan(48)] tan(41)  =

[tan(48)tan(41)] / [tan(41) + tan(48)] * 140 = about 68.2669m or 68.27m or just 68.3m  (rounded to nearest tenth)......which is pretty much what heureka found!!!

Here's a picture to demonstrate the situation:

Why so complicated ?

h/tan(41) + h/tan(48) = 140,

x isn't needed.

Thanks, Anonymous....genius in simplicity...!!!!

Melody: Hi Melody, and likewise!  I appreciate the explanation, and as for more help, it isn't necessary thanks to CPhills answer. 

CPhill: Hi, thanks for the detailed explanation. Definitely makes sense. And that was nice of you to take the time to create a picture of the situation. 

heureka: I'm a bit confused by your use of sin rather than tan...? Wouldn't that result in a totally different answer? Yet, you have arrived at the correct answer.