A number minus its reciprocal is 3.75. Find the number.

Let the number be $x$.
Its reciprocal is $\frac{1}{x}$.
The difference between the number and its reciprocal is 3.75.

$$x - \frac{1}{x} = 3.75$$

$$3.75 = \frac{375}{100} = \frac{15}{4}$$

So the equation becomes:

$$x - \frac{1}{x} = \frac{15}{4}$$

$$4x \left(x - \frac{1}{x}\right) = 4x \left(\frac{15}{4}\right)$$ $$4x^2 - 4 = 15x$$

We look for two numbers that multiply to $4 \times (-4) = -16$ and add up to -15. These numbers are -16 and 1.

$$4x^2 - 16x + x - 4 = 0$$

Factor by grouping:

$$4x(x - 4) + 1(x - 4) = 0$$ $$(4x + 1)(x - 4) = 0$$

Set each factor to zero:

$$4x + 1 = 0 \implies 4x = -1 \implies x = -\frac{1}{4}$$ $$x - 4 = 0 \implies x = 4$$

The number can be either 4 or $-\frac{1}{4}$.

The number is $\boxed{4 \text{ or } -\frac{1}{4}}$.

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Area ($A$) = $7462 \text{ cm}^2$
Width ($W$) = $82 \text{ cm}$
Area formula: $A = L \times W$
We need to find the length ($L$).

$$L = \frac{A}{W} = \frac{7462}{82}$$

$$\log L = \log\left(\frac{7462}{82}\right)$$ $$\log L = \log(7462) - \log(82)$$

The characteristic for 7462 (a 4-digit number) is $4-1=3$.
From the logarithm table, the mantissa for 7462 (looking up 74, then 6, then mean difference for 2) is approximately $0.8727 + 0.0001 = 0.8728$.

$$\log(7462) = 3.8728$$

The characteristic for 82 (a 2-digit number) is $2-1=1$.
From the logarithm table, the mantissa for 82 (looking up 82, then 0) is approximately 0.9138.

$$\log(82) = 1.9138$$

$$\log L = 3.8728 - 1.9138$$ $$\log L = 1.9590$$

The characteristic is 1, which means the number has $1+1=2$ digits before the decimal point.
The mantissa is 0.9590.
From the antilogarithm table, the number corresponding to the mantissa 0.9590 is approximately 9.096.
Therefore, $L = 9.096 \times 10^1$.

$$L = 90.96 \text{ cm}$$

The length of the farm is $\boxed{90.96 \text{ cm}}$.