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Simple Calculus Problem (I hope)

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Maetryx

Maetryx

Diamond Member
This should be relatively easy, but I have not been in a caculus class for a long time. I believe this is a simple calculus problem.

I need a formula that will give the following results:

Tree trunk at breast height has a diameter (dbh) of 4 inches.
Tree has a canopy of 8 feet.

Bigger Tree has a dbh of 20 inches.
Bigger Tree has a canopy of 20 feet.

What is the function of dbh that yields these results? f(dbh) = canopy

You can also assume that 0 inches dbh yields 0 feet canopy.
 
You have three points. (4,8), (20,20), and (0,0)

Since the points aren't collinear, you could create a quadratic (parabola) that contains all three points, or any higher ordered polynomial.

What kind of model do you think this is? What kind of assumptions can you make?

There's no "the" function; there are limitless functions that would provide those results. Of course, most of those, while fitting those 3 data points, might not make much sense for other values.


edit: incidentally, just a wild guess by me, but I'd think something like a logistics growth curve would probably work best. Horizontal branches really can only stick out so far from the center before they would eventually succumb to external forces (gravity, wind.) Thus, there's probably a limit to the canopy size, while the trunk could continue to grow.
 
dbh is the the diameter of the tree about 4 feet above the ground, and for whatever reason landscape architects decided to call it "breast height". Apparently, elderly women are discriminated against by landscape architects. 🙂

Okay, I realize there is no "THE EQUATION", so that's a fair place to start. I'm hoping for a polynomial with the highest order being 2 (if I'm saying this right). In fact, I think we may find that

canopy = f(dbh) = a*(dbh)^2 + b(dbh) + c [quadratic]

where:

Canopy is the diameter of the of the tree idealized as a circle when looking down from an aerial view;
And Canopy is a function of the trunk diameter dbh;
And Canopy is correlated to the square of dbh.
And a, b and c are constants that makes my equation work.

Since we have the point 0,0 as a valid point on the function, thenc should be zero.
 
Originally posted by: Maetryx
dbh is the the diameter of the tree about 4 feet above the ground, and for whatever reason landscape architects decided to call it "breast height". Apparently, elderly women are discriminated against by landscape architects. 🙂

Okay, I realize there is no "THE EQUATION", so that's a fair place to start. I'm hoping for a polynomial with the highest order being 2 (if I'm saying this right). In fact, I think we may find that

canopy = f(dbh) = a*(dbh)^2 + b(dbh) + c [quadratic]

where:

Canopy is the diameter of the of the tree idealized as a circle when looking down from an aerial view;
And Canopy is a function of the trunk diameter dbh;
And Canopy is correlated to the square of dbh.
And a, b and c are constants that makes my equation work.

Since we have the point 0,0 as a valid point on the function, thenc should be zero.
Click to expand...

You have two other sets of data points, and two unknowns. If you plug in the values you have from the two trees, that will give you two equations, each with the unknowns a and b and no other unknowns. That would be enough information to solve for a and b.

Do be careful about units. It is easiest to convert everything to inches.

Also, tee hee.
 
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