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resuelve el siguiente problema de limites y continuidad 

The problem in the image deals with the "Doyle log rule," a method to estimate lumber yield from a log, based on its diameter \( d \) (in inches) and its length \( L \) (in feet). The formula provided is:

\[
N(d, L) = \left(\frac{d - 4}{4}\right)^2 L
\]

### Part (a): 
We are asked to find the number of board-feet of lumber from a log with a diameter of 22 inches and a length of 12 feet.

Substitute \( d = 22 \) and \( L = 12 \) into the formula:

\[
N(22, 12) = \left(\frac{22 - 4}{4}\right)^2 \times 12
\]
\[
N(22, 12) = \left(\frac{18}{4}\right)^2 \times 12
\]
\[
N(22, 12) = \left(4.5\right)^2 \times 12
\]
\[
N(22, 12) = 20.25 \times 12 = 243 \text{ board-feet}
\]

So, the yield is **243 board-feet**.

### Part (b):
We are asked to evaluate \( N(30, 12) \).

Substitute \( d = 30 \) and \( L = 12 \) into the formula:

\[
N(30, 12) = \left(\frac{30 - 4}{4}\right)^2 \times 12
\]
\[
N(30, 12) = \left(\frac{26}{4}\right)^2 \times 12
\]
\[
N(30, 12) = \left(6.5\right)^2 \times 12
\]
\[
N(30, 12) = 42.25 \times 12 = 507 \text{ board-feet}
\]

Thus, \( N(30, 12) = 507 \) board-feet.

### Final Answers:
- (a) \( N(22, 12) = 243 \)
- (b) \( N(30, 12) = 507 \)

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Would you like more details or have any questions about these calculations?

Here are some related questions for further practice:
1. What happens to the number of board-feet as the diameter increases?
2. How would the formula change if the length of the log was doubled?
3. Evaluate \( N(25, 10) \) using the Doyle log rule.
4. If the diameter was 40 inches and the length was 20 feet, how many board-feet of lumber would be yielded?
5. How does the Doyle rule compare to other log rules like the Scribner or International rule?

**Tip:** The Doyle log rule tends to underestimate the yield for smaller logs, while it provides more accurate estimates for larger logs.

La regla de los troncos de Doyle es uno de varios métodos para determinar el rendimiento en madera aserrada (en tablones-pie) en términos de su diámetro (en pulgadas) y su longitud (en pies). Hallar el número de tablones-pie para un tronco de 22 pulgadas de diámetro y 12 pies de longitud. Evaluar también para un diámetro de 30 pulgadas y 12 pies de longitud.

Learn how to use the Doyle log rule to calculate lumber yield from logs of various diameters and lengths. This example covers logs with diameters of 22 inches and 30 inches, with a length of 12 feet. Key mathematical concepts include applied geometry and algebra.

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