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

**Problem 38**

A log 18 ft. long is 2 ft. in diameter at the top end and 3 ft. in diameter at the butt end.

- How many cubic feet of wood does the log contain?
- How many cubic feet are there in the largest piece of timber of square cross section that can be cut from the log?
- How many cubic feet are in the largest piece of square timber of the same size throughout its whole length?
- How many board feet does the piece of timber in (c), a board foot being equivalent to a board 1 ft. square and 1 in. thick?

Hint: In (b) the larger end is the square ABCD. What is the smaller end? In (c) one end is the square EFGH. What is the other end?

**Solution 38**

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**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*

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