The Doyle Log Rule is a formula that gives the lumber volume V, in board feet, that can be obtained from a log of diameter D inches and length L feet.+ V = (D – 4)22 V=> 16 Here the diameter is measured at the small end, and we assume that D > 4. If we let E = D – 4, this formula can be written as the following v=1 Suppose we are dealing with 16-foot-long logs. (a) Express V in terms of E. V = (b) If the value of E for one log is twice that of another log, how do the amounts of lumber compare? The log with the larger value of E produces times as much lumber. (c) How are the diameters of the two logs from part (b) related? To find the diameter of the log with the larger value of E, we multiply the diameter of the other log by and then subtract inches.
The Correct Answer and Explanation is :
(a) Expressing V in terms of E:
The Doyle Log Rule is given by:
[
V = \frac{(D – 4)^2 L}{16}
]
where (V) is the volume in board feet, (D) is the diameter of the log in inches (at the small end), and (L) is the length of the log in feet. In this case, the length (L = 16) feet.
Let (E = D – 4), so we can substitute (E) for (D – 4):
[
V = \frac{E^2 \cdot 16}{16}
]
The 16’s cancel out, so the volume (V) in terms of (E) becomes:
[
V = E^2
]
(b) Comparing the amounts of lumber when (E) is doubled:
Suppose the value of (E) for one log is twice that of another log. Let the first log have an (E) value of (E_1), and the second log have an (E) value of (E_2 = 2E_1).
The volume of lumber for the first log is:
[
V_1 = E_1^2
]
The volume of lumber for the second log is:
[
V_2 = E_2^2 = (2E_1)^2 = 4E_1^2
]
So, the log with the larger (E) (i.e., the second log) produces four times as much lumber as the first log.
(c) Relating the diameters of the two logs:
Since (E = D – 4), the diameter (D) is related to (E) by (D = E + 4).
For the first log, the diameter is (D_1 = E_1 + 4), and for the second log, the diameter is (D_2 = E_2 + 4 = 2E_1 + 4).
So, the diameter of the second log is related to the diameter of the first log by:
[
D_2 = 2D_1 – 4
]
This means that the diameter of the second log is twice the diameter of the first log, minus 4 inches.
Summary of Answers:
- (a) The volume (V) in terms of (E) is (V = E^2).
- (b) The log with the larger value of (E) produces 4 times as much lumber.
- (c) The diameter of the log with the larger (E) is twice the diameter of the other log, minus 4 inches.