038 Review Problem - Circular log with non-uniform cross-section

Part (a): Volume of log
$V = \frac{1}{3}(A_1 + A_2 + \sqrt{A_1 \, A_2}) \, h = \frac{1}{3}\pi (R^2 + r^2 + Rr) \, h$

$V = \frac{1}{3}\pi [ \, 1.5^2 + 1.0^2 + 1.5(1.0) \, ] \, (18)$

$V = 28.5\pi ~ \text{ft.}^3 = 89.54 ~ \text{ft.}^3$           answer
 

Part (b): Volume of the largest timber of square cross-section
$a^2 + a^2 = 3^2$

$2a^2 = 9$

$a^2 = 4.5$
 

$b^2 + b^2 = 2^2$

$2b^2 = 4$

$b^2 = 2$
 

$A_1 = a^2 = 4.5 ~ \text{ft.}^2$

$A_2 = b^2 = 2 ~ \text{ft.}^2$
 

$V = \frac{1}{3}(A_1 + A_2 + \sqrt{A_1 \, A_2 }) \, h$

$V = \frac{1}{3}(4.5 + 2 + \sqrt{4.5 \cdot 2})(18)$

$V = 57 ~ \text{ft.}^3$           answer
 

Part (c): Volume of largest square timber of the same size throughout its length:
$V = A_2 h = 2(18) = 36 ~ \text{ft.}^3$           answer
 

Part (d): Number of board feet of timber in (c)
$V = 36 ~ \text{ft.}^3 \times \dfrac{1 ~ \text{bd. ft.}}{1 ~ \text{ft.} \times 1 ~ \text{ft.} \times \frac{1}{12} ~ \text{ft.}}$

$V = 432 ~ \text{bd. ft.}$           answer