Determine the kinetic energy of the object as a function of time.

The mass of the object is $m = 4.0$ kg.
The speed of the object is given by $v = (2t)$ m/s.
The kinetic energy ($K$) is given by the formula $K = \frac{1}{2}mv^2$.
Substitute the given values:

$$K = \frac{1}{2}(4.0 \, \text{kg})(2t)^2$$ $$K = \frac{1}{2}(4.0 \, \text{kg})(4t^2 \, \text{m}^2/\text{s}^2)$$ $$K = (2.0)(4t^2) \, \text{J}$$ $$K = 8.0t^2 \, \text{J}$$

Power ($P$) is the rate of change of kinetic energy: $P = \frac{dK}{dt}$.

$$P = \frac{d}{dt}(8.0t^2)$$ $$P = 8.0 \times 2t$$ $$P = 16.0t \, \text{W}$$

Substitute $t = 1$ s into the power equation:

$$P = 16.0(1) \, \text{W}$$ $$P = 16 \, \text{W}$$

Alternatively, using $P = Fv$:

Given $v = 2t$, the acceleration $a = \frac{dv}{dt}$.

$$a = \frac{d}{dt}(2t) = 2 \, \text{m/s}^2$$

Using Newton's second law, $F = ma$.

$$F = (4.0 \, \text{kg})(2 \, \text{m/s}^2)$$ $$F = 8.0 \, \text{N}$$

Since the force causes the acceleration in the direction of motion, the force and velocity are in the same direction.

$$P = Fv$$ $$P = (8.0 \, \text{N})(2 \, \text{m/s})$$ $$P = 16 \, \text{W}$$

Both methods yield the same result.

The final answer is $\text{b. 16 W}$.