A) Step 1: Set up the differential equation and its solution.

Given $M_0 = 50$ kg.
At $t=5$ hours, $M(5) = 20$ kg.
Substitute these values into the equation:

Divide by 50:

Take the natural logarithm of both sides:

If 90% of the sugar is dissolved, then 10% remains.
Remaining amount $= 0.10 \times M_0 = 0.10 \times 50 \text{ kg} = 5 \text{ kg}$.

We need to find how much longer it will take from $t=5$ hours until the amount of sugar remaining is 5 kg. Let this additional time be $T_{add}$.
The amount of sugar at time $t = 5 + T_{add}$ is $M(5+T_{add}) = 5$ kg.
Using the formula $M(t) = M_0 e^{-kt}$:

We know from Step 2 that $50 e^{-5k} = 20$. Substitute this into the equation:

Divide by 20:

Take the natural logarithm of both sides:

Now substitute the value of $k = \frac{\ln(2.5)}{5}$:

Using a calculator:

It will take approximately $\boxed{\text{7.564 hours}}$ longer.

Combining $x \ge 0$ and $x \ne 0$, we get $x > 0$.
The cosine function is defined for all real numbers, so there are no further restrictions from the cosine itself.
Thus, the domain of definition is all positive real numbers.
The domain of definition is $\boxed{(0, \infty)}$.

Recall that the sine function is an odd function, meaning $\sin(-\theta) = -\sin(\theta)$.
Applying this identity:

Factor out $-1$ from $f(-x)$:

Since $f(x) = x + \sin 4x$, we can see that:

Therefore, the function $f(x) = x + \sin 4x$ is an odd function.

The function is $\boxed{\text{odd}}$.