A board foot is a unit for measuring wood. One board foot has a volume of 144 cubic inches. The Doyle log rule, given by b=l(r-2/2)^2, is a formula for approximating the number b of board feet in a log with length l (in feet) and radius r (in inches). The total volume V (in cubic inches) of wood in the main trunk of a Douglas fir can be modeled by V=250^3 where r is the radius of the trunk at the base of the tree. Suppose you need 6000 board feet from a 40 foot Douglas fir log. a) What volume of wood do you need?
b) What is the radius of a log that will meet your needs?
c) What is the total volume of wood in the main trunk of a Douglas fir that will meet your needs?
d) If you find a suitable tree, what fraction of the tree would you actually use?
e) How does your answer in part d change if you instead need only 2000 board feet?