Height Calculation Example  

TOPIC: Will's new height champion

== 1 of 7 ==
Date: Thurs, Jan 17 2008 9:05 am
From: dbhguru@comcast.net

In this e-mail, I will present a problem that our distinguished president Will Blozan might conceivably face while climbing one of his magnificent Smoky Mountain hemlocks. We'll solve the problem in this e-mail. Here goes.
It is a sunny, still day and Will is high in a hemlock taking circumferential measurements of a huge hemlock that he believes will set a new volume record for the species. As he approaches the top, he cranes his neck around and suddenly spies what appear to be an even larger, taller hemlock some distance away. The tree is humungus. Where had it been all the time, Will wonders? Its base is lower than the one he is in and its top appears higher. Will yells HOLY MOLY in honor of Dale. He gets so excited that he soils himself.
Will just has to get some measurements of this new tree, but in reaching for his laser, he discovers that it is nowhere to be found. All he has is his tape measure and clinometer. Ah yes, and he has his cell phone. He quickly dials up his buddy Bob in Massachusetts who happens to be home nursing his aches and pains. Will is so excited that he has trouble explaining what he is seeing, but the message finally gets across and Bob also gets excited and he too soils himself.
In due course, the two amidst the unpleasnt odor that each exudes settles down to the task at hand. Will asks Bob if he can generate a handy dandy formula or something. Bob says sure and waddles carefully over to his computer. Bob asks if Will can shoot the angle over to the top of the super hemlock, which appears to stick above the top of the one Will is in. Will calls back a positive angle of a1 to the top of the super tree. Bob writes down the angle and then asks Will if he can climb down a measured distance and shoot a second angle to the top of the super hemlock? Will complies and calls back a vertical distance descended of d and a second angle across to the top of Mr. Super Tree of a2. Now since Will was shooting up to the crown of the super tree from his upper vantage point, a level line across from the top of the super tree would ride above the top of the tree Will is in. Nonetheless we can form a triangle from Will's eye to the top of the super tree then level back to inte
rstect the extension of the top of the tree Will is in and then down to Will's eye 9when at the upper point). We will call the distance from the imaginary point of intersection down to Will's eye as d0. We are assuming the two trees are vertical to their tips for simplicity's sake.

We now have the means of establishing two equations:

x = tan(a2)*(d+d0)

x = tan(90-a1)*(d0)

where x is the level distance between the vertical lines formed by the two trunks (we've assumed verticality). The two simultaneous equations can be set equal to one another and solved for d0.

tan(a2)*(d+d0) = tan(90-a1)*(d0)

expanding the left side, we get

tan(a2)*d + tan(a2)*d0 = tan(90-a1)*(d0)

algebraically rearranging, we get

[tan(90-a1) - tan(a2)]*(d0) = tan(a2)*d

 d0 = [tan(a2)*d]/[tan(90-a1) - tan(a2)]

We define D = d + do

Then, we can compute x as

x = tan(a2)*D, the level distance between the two trunks.

From Will's lower vantage point, we can compute the amount of the super tree's height above Will's eye as

h1 = tan(90-a2)*x

Will then descends his tree, measuring its remaining height down to its base as h2
Will then shoots a point level on the trunk of the super tree from the elevetion of the base of his tree and goes over to the super tree and measures from that level point to its base. He gets h3.
Will proudly proclaims the height of the super tree as H = h1+h2+h3, a new world record!
Monica enters the basement, takes a whiff and orders Bob to immediately hit the shower. Will has to wait. Whew! What we do to crown new champions.