Given data:
Current density (J) = 20 sin(x)i + y cos(z)j at the origin (0, 0, 0)

To find the charge density (ρ), we can use the continuity equation, which states that the divergence of the current density is equal to the negative rate of change of charge density with respect to time.

Mathematically, this can be expressed as:
∇ · J = - ∂ρ/∂t

Since the current density is given in Cartesian coordinates, we can write the divergence of J as:
∇ · J = (∂(20 sin(x)i)/∂x) + (∂(y cos(z)j)/∂y)

Calculating the partial derivatives, we get:
∇ · J = (20 cos(x)i) + (cos(z)j)

Since we are interested in finding the charge density at the origin (0, 0, 0), we substitute the values into the expression:
∇ · J = (20 cos(0)i) + (cos(0)j)
∇ · J = (20i) + (j)

From the continuity equation, we know that the divergence of the current density (∇ · J) is equal to the negative rate of change of charge density (∂ρ/∂t). Since there is no information given about the time derivative (∂ρ/∂t), we can assume it to be zero or negligible.

Therefore, we can equate the divergence of the current density (∇ · J) to zero:
(20i) + (j) = 0

Comparing the coefficients of the unit vectors, we get:
20i + j = 0

This implies that the charge density at the origin is 0.

Hence, none of the given options (a) 20t, (b) 21t, (c) 19t, or (d) -20t is correct.

Therefore, the correct answer is none of the provided options.