Question

The number of board feet in a 16 foot long tree is approximated by the model F(d) = 0.77d^2 -- 1.32d -- 9.31 where F is the number of board feet and d is the diameter (in inches) of the log. What diameter tree will produce the minimum number of board feet?

Answer 1

we are given

$F(d)=0.77d^2-1.32d-9.31$

where

F is the number of board feet

d is the diameter (in inches) of the log

Since, we have to minimize F

so, firstly we will find it's derivative

$F'(d)=0.77\times 2d-1.32\times 1-0$

$F'(d)=1.54d-1.32$

now, we can set it to 0

and then we can solve for d

$F'(d)=1.54d-1.32=0$

$1.54d-1.32=0$

so, we get diameter as

$d=\frac{6}{7}inch$..............Answer