Question

Please help.

The volume of wood contained in the trunk of a particular kind if for tree is modeled by: V=0.002198d^1.739925h^1.123187 in cubic feet, where d is the diameter if the tree and h is the height of the tree. If the diameter of the tree is 10 inches, how quickly is the volume of the wood changing when the tree is 32 feet and it's height is increasing by half a foot per year. (The diameter of the tree remains constant. )*

Answer 1

Answer: 0.0010286521



Solution:


My assumption:
By: V=0.002198d^1.739925h^1.123187V=0.002198d^1.739925h^1.123187
You mean: V=0.002198d*1.739925h*1.123187


Step 1: convert diameter from inches to feet (because the formulate is using cubic feet)


diameter: 10 inches * (1 foot/12 inches)=0.83 feet



Step 2: Since diameter is constant rewrite original equation as:


V=0.002198 (0.83) * 1.739925h * 1.123187
=0.001824 * 1.79925h * 1.123187


Step 3:



Since the tree 32 feet during year 1, by year 2 its height would be 32.5, year 3 it would 33 feet, and so on...


Step 4:


Calculate yearly volume, V, by plugging in yearly height values


Step 5:


Calculate the yearly rate of change by subtracting volume of year 1 (V1) from volume of year 2 (V2): The rate is
(V2-V1)/year=0.001028652


* see table below
Year Diameter (d) height (h) constant Volume Rate
1 0.0018316667 32 1.123187 0.065833734
2 0.0018316667 32.5 1.123187 0.0668623861 0.0010286521
3 0.0018316667 33 1.123187 0.0678910382 0.0010286521
4 0.0018316667 33.5 1.123187 0.0689196903 0.0010286521
5 0.0018316667 34 1.123187 0.0699483424 0.0010286521


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